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Properties

Label 2-334-167.100-c1-0-0
Degree 22
Conductor 334334
Sign −0.970−0.241i-0.970 - 0.241i
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  + (−0.726 + 0.686i)2-s + (0.683 + 3.23i)3-s + (0.0567 − 0.998i)4-s + (−0.0475 − 2.51i)5-s + (−2.72 − 1.88i)6-s + (1.57 + 2.01i)7-s + (0.644 + 0.764i)8-s + (−7.27 + 3.21i)9-s + (1.76 + 1.79i)10-s + (−1.95 + 1.14i)11-s + (3.27 − 0.499i)12-s + (−6.48 + 2.58i)13-s + (−2.52 − 0.385i)14-s + (8.10 − 1.87i)15-s + (−0.993 − 0.113i)16-s + (4.71 + 3.00i)17-s + ⋯
L(s)  = 1  + (−0.513 + 0.485i)2-s + (0.394 + 1.86i)3-s + (0.0283 − 0.499i)4-s + (−0.0212 − 1.12i)5-s + (−1.11 − 0.769i)6-s + (0.594 + 0.762i)7-s + (0.227 + 0.270i)8-s + (−2.42 + 1.07i)9-s + (0.556 + 0.567i)10-s + (−0.587 + 0.344i)11-s + (0.944 − 0.144i)12-s + (−1.79 + 0.715i)13-s + (−0.675 − 0.103i)14-s + (2.09 − 0.483i)15-s + (−0.248 − 0.0283i)16-s + (1.14 + 0.728i)17-s + ⋯

Functional equation

Λ(s)=(334s/2 ΓC(s) L(s)=((−0.970−0.241i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(334s/2 ΓC(s+1/2) L(s)=((−0.970−0.241i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 334334    =    2⋅1672 \cdot 167
Sign: −0.970−0.241i-0.970 - 0.241i
Analytic conductor: 2.667002.66700
Root analytic conductor: 1.633091.63309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ334(267,⋅)\chi_{334} (267, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 334, ( :1/2), −0.970−0.241i)(2,\ 334,\ (\ :1/2),\ -0.970 - 0.241i)

Particular Values

L(1)L(1) ≈\approx 0.117696+0.961261i0.117696 + 0.961261i
L(12)L(\frac12) ≈\approx 0.117696+0.961261i0.117696 + 0.961261i
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+(0.726−0.686i)T 1 + (0.726 - 0.686i)T
167 1+(−1.61+12.8i)T 1 + (-1.61 + 12.8i)T
good3 1+(−0.683−3.23i)T+(−2.74+1.21i)T2 1 + (-0.683 - 3.23i)T + (-2.74 + 1.21i)T^{2}
5 1+(0.0475+2.51i)T+(−4.99+0.189i)T2 1 + (0.0475 + 2.51i)T + (-4.99 + 0.189i)T^{2}
7 1+(−1.57−2.01i)T+(−1.70+6.78i)T2 1 + (-1.57 - 2.01i)T + (-1.70 + 6.78i)T^{2}
11 1+(1.95−1.14i)T+(5.37−9.59i)T2 1 + (1.95 - 1.14i)T + (5.37 - 9.59i)T^{2}
13 1+(6.48−2.58i)T+(9.44−8.92i)T2 1 + (6.48 - 2.58i)T + (9.44 - 8.92i)T^{2}
17 1+(−4.71−3.00i)T+(7.16+15.4i)T2 1 + (-4.71 - 3.00i)T + (7.16 + 15.4i)T^{2}
19 1+(1.84−3.60i)T+(−11.1−15.4i)T2 1 + (1.84 - 3.60i)T + (-11.1 - 15.4i)T^{2}
23 1+(−6.11+1.90i)T+(18.9−13.0i)T2 1 + (-6.11 + 1.90i)T + (18.9 - 13.0i)T^{2}
29 1+(2.83+0.886i)T+(23.8+16.5i)T2 1 + (2.83 + 0.886i)T + (23.8 + 16.5i)T^{2}
31 1+(0.542+4.07i)T+(−29.9+8.11i)T2 1 + (0.542 + 4.07i)T + (-29.9 + 8.11i)T^{2}
37 1+(−6.93−3.06i)T+(24.8+27.3i)T2 1 + (-6.93 - 3.06i)T + (24.8 + 27.3i)T^{2}
41 1+(−3.17+0.862i)T+(35.3−20.7i)T2 1 + (-3.17 + 0.862i)T + (35.3 - 20.7i)T^{2}
43 1+(2.24−0.170i)T+(42.5−6.48i)T2 1 + (2.24 - 0.170i)T + (42.5 - 6.48i)T^{2}
47 1+(−9.41−7.06i)T+(13.1+45.1i)T2 1 + (-9.41 - 7.06i)T + (13.1 + 45.1i)T^{2}
53 1+(−3.42+1.83i)T+(29.3−44.1i)T2 1 + (-3.42 + 1.83i)T + (29.3 - 44.1i)T^{2}
59 1+(−2.53+1.62i)T+(24.8−53.4i)T2 1 + (-2.53 + 1.62i)T + (24.8 - 53.4i)T^{2}
61 1+(0.290−0.871i)T+(−48.8−36.5i)T2 1 + (0.290 - 0.871i)T + (-48.8 - 36.5i)T^{2}
67 1+(−0.145+7.70i)T+(−66.9−2.53i)T2 1 + (-0.145 + 7.70i)T + (-66.9 - 2.53i)T^{2}
71 1+(−8.39−7.34i)T+(9.37+70.3i)T2 1 + (-8.39 - 7.34i)T + (9.37 + 70.3i)T^{2}
73 1+(10.8−1.23i)T+(71.1−16.4i)T2 1 + (10.8 - 1.23i)T + (71.1 - 16.4i)T^{2}
79 1+(−10.5−3.74i)T+(61.3+49.7i)T2 1 + (-10.5 - 3.74i)T + (61.3 + 49.7i)T^{2}
83 1+(−0.129−0.122i)T+(4.70+82.8i)T2 1 + (-0.129 - 0.122i)T + (4.70 + 82.8i)T^{2}
89 1+(4.00+0.925i)T+(79.9+39.0i)T2 1 + (4.00 + 0.925i)T + (79.9 + 39.0i)T^{2}
97 1+(0.0717−0.538i)T+(−93.6−25.4i)T2 1 + (0.0717 - 0.538i)T + (-93.6 - 25.4i)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

−11.81219294406015274817134452304, −10.70394233904279951485171141592, −9.781253071312958745097173202030, −9.311285824001888704237874816353, −8.460355498073946553654506970300, −7.75170928814285074720956570515, −5.63419879398033619281149082918, −5.01989259000747921543548596410, −4.28200351455711434205113593468, −2.43124678868022497080853072641, 0.76433168766087696457038781384, 2.47363108958515855469423608181, 3.04825018086542026393477338866, 5.35449820088457539773582238673, 6.96824958419846274374362390760, 7.37693110404926727869280614670, 7.87997071577110781480874019446, 9.166682452790109260314183168077, 10.44564494651848378140674576490, 11.18372150358776302645486052478

Graph of the ZZ-function along the critical line