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Properties

Label 2-334-167.107-c1-0-10
Degree 22
Conductor 334334
Sign −0.992−0.126i-0.992 - 0.126i
Analytic cond. 2.667002.66700
Root an. cond. 1.633091.63309
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.862 − 0.505i)2-s + (−0.996 − 0.311i)3-s + (0.489 + 0.872i)4-s + (−0.305 + 1.77i)5-s + (0.702 + 0.772i)6-s + (0.166 + 0.498i)7-s + (0.0189 − 0.999i)8-s + (−1.57 − 1.08i)9-s + (1.16 − 1.38i)10-s + (0.0502 − 0.884i)11-s + (−0.215 − 1.02i)12-s + (−4.08 − 1.10i)13-s + (0.108 − 0.514i)14-s + (0.858 − 1.67i)15-s + (−0.521 + 0.853i)16-s + (−1.27 − 3.03i)17-s + ⋯
L(s)  = 1  + (−0.610 − 0.357i)2-s + (−0.575 − 0.179i)3-s + (0.244 + 0.436i)4-s + (−0.136 + 0.795i)5-s + (0.286 + 0.315i)6-s + (0.0628 + 0.188i)7-s + (0.00669 − 0.353i)8-s + (−0.523 − 0.362i)9-s + (0.367 − 0.436i)10-s + (0.0151 − 0.266i)11-s + (−0.0622 − 0.294i)12-s + (−1.13 − 0.307i)13-s + (0.0290 − 0.137i)14-s + (0.221 − 0.433i)15-s + (−0.130 + 0.213i)16-s + (−0.308 − 0.735i)17-s + ⋯

Functional equation

Λ(s)=(334s/2 ΓC(s) L(s)=((−0.992−0.126i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(334s/2 ΓC(s+1/2) L(s)=((−0.992−0.126i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 334334    =    2⋅1672 \cdot 167
Sign: −0.992−0.126i-0.992 - 0.126i
Analytic conductor: 2.667002.66700
Root analytic conductor: 1.633091.63309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ334(107,⋅)\chi_{334} (107, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 334, ( :1/2), −0.992−0.126i)(2,\ 334,\ (\ :1/2),\ -0.992 - 0.126i)

Particular Values

L(1)L(1) ≈\approx 0.00251495+0.0396994i0.00251495 + 0.0396994i
L(12)L(\frac12) ≈\approx 0.00251495+0.0396994i0.00251495 + 0.0396994i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.862+0.505i)T 1 + (0.862 + 0.505i)T
167 1+(12.0+4.57i)T 1 + (12.0 + 4.57i)T
good3 1+(0.996+0.311i)T+(2.46+1.70i)T2 1 + (0.996 + 0.311i)T + (2.46 + 1.70i)T^{2}
5 1+(0.305−1.77i)T+(−4.71−1.67i)T2 1 + (0.305 - 1.77i)T + (-4.71 - 1.67i)T^{2}
7 1+(−0.166−0.498i)T+(−5.60+4.19i)T2 1 + (-0.166 - 0.498i)T + (-5.60 + 4.19i)T^{2}
11 1+(−0.0502+0.884i)T+(−10.9−1.24i)T2 1 + (-0.0502 + 0.884i)T + (-10.9 - 1.24i)T^{2}
13 1+(4.08+1.10i)T+(11.2+6.57i)T2 1 + (4.08 + 1.10i)T + (11.2 + 6.57i)T^{2}
17 1+(1.27+3.03i)T+(−11.9+12.1i)T2 1 + (1.27 + 3.03i)T + (-11.9 + 12.1i)T^{2}
19 1+(1.68+0.822i)T+(11.6+14.9i)T2 1 + (1.68 + 0.822i)T + (11.6 + 14.9i)T^{2}
23 1+(7.95−3.51i)T+(15.4−17.0i)T2 1 + (7.95 - 3.51i)T + (15.4 - 17.0i)T^{2}
29 1+(8.07+3.57i)T+(19.5+21.4i)T2 1 + (8.07 + 3.57i)T + (19.5 + 21.4i)T^{2}
31 1+(−1.58+0.630i)T+(22.5−21.2i)T2 1 + (-1.58 + 0.630i)T + (22.5 - 21.2i)T^{2}
37 1+(5.84−4.04i)T+(13.0−34.6i)T2 1 + (5.84 - 4.04i)T + (13.0 - 34.6i)T^{2}
41 1+(2.49−2.36i)T+(2.32−40.9i)T2 1 + (2.49 - 2.36i)T + (2.32 - 40.9i)T^{2}
43 1+(−2.92−2.37i)T+(8.88+42.0i)T2 1 + (-2.92 - 2.37i)T + (8.88 + 42.0i)T^{2}
47 1+(4.03+2.16i)T+(26.0+39.1i)T2 1 + (4.03 + 2.16i)T + (26.0 + 39.1i)T^{2}
53 1+(2.66+9.13i)T+(−44.6+28.5i)T2 1 + (2.66 + 9.13i)T + (-44.6 + 28.5i)T^{2}
59 1+(2.15−5.13i)T+(−41.3−42.1i)T2 1 + (2.15 - 5.13i)T + (-41.3 - 42.1i)T^{2}
61 1+(−2.77+11.0i)T+(−53.7−28.8i)T2 1 + (-2.77 + 11.0i)T + (-53.7 - 28.8i)T^{2}
67 1+(−1.70−9.89i)T+(−63.1+22.3i)T2 1 + (-1.70 - 9.89i)T + (-63.1 + 22.3i)T^{2}
71 1+(−4.10+0.786i)T+(65.9−26.2i)T2 1 + (-4.10 + 0.786i)T + (65.9 - 26.2i)T^{2}
73 1+(1.93+3.16i)T+(−33.2+64.9i)T2 1 + (1.93 + 3.16i)T + (-33.2 + 64.9i)T^{2}
79 1+(−15.4−1.17i)T+(78.0+11.9i)T2 1 + (-15.4 - 1.17i)T + (78.0 + 11.9i)T^{2}
83 1+(8.07−4.73i)T+(40.5−72.3i)T2 1 + (8.07 - 4.73i)T + (40.5 - 72.3i)T^{2}
89 1+(−1.50−2.94i)T+(−52.0+72.2i)T2 1 + (-1.50 - 2.94i)T + (-52.0 + 72.2i)T^{2}
97 1+(−4.64−1.84i)T+(70.5+66.6i)T2 1 + (-4.64 - 1.84i)T + (70.5 + 66.6i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−11.26472602460600254317472543951, −10.18905223258397344085568972408, −9.414261878584136800945875439842, −8.236516005311753412634972525860, −7.25739398871403276452922282008, −6.37375871410145265421826092340, −5.22611383162453156588560036110, −3.52360285298388318265375745916, −2.31743121922312073954223488370, −0.03371578345849030252214087909, 2.06287499936268107195555854037, 4.26687094781197803153387635316, 5.22545943895526725421416042804, 6.20499247872842008968300079249, 7.41097086341672710590512977546, 8.342132061352627075835891439226, 9.155143207898469318648447677775, 10.25425433208388023271845384359, 10.90160201960093013068423721525, 12.07654947331746046688662961430

Graph of the ZZ-function along the critical line