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Properties

Label 2-334-167.108-c1-0-4
Degree 22
Conductor 334334
Sign 0.910+0.412i0.910 + 0.412i
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  + (0.752 + 0.658i)2-s + (−3.27 + 0.124i)3-s + (0.132 + 0.991i)4-s + (−1.74 − 1.11i)5-s + (−2.54 − 2.06i)6-s + (0.0421 + 0.0689i)7-s + (−0.553 + 0.832i)8-s + (7.74 − 0.587i)9-s + (−0.581 − 1.99i)10-s + (1.96 + 0.376i)11-s + (−0.556 − 3.23i)12-s + (1.36 − 3.63i)13-s + (−0.0136 + 0.0795i)14-s + (5.87 + 3.44i)15-s + (−0.965 + 0.261i)16-s + (1.43 − 5.72i)17-s + ⋯
L(s)  = 1  + (0.531 + 0.465i)2-s + (−1.89 + 0.0716i)3-s + (0.0660 + 0.495i)4-s + (−0.782 − 0.498i)5-s + (−1.04 − 0.843i)6-s + (0.0159 + 0.0260i)7-s + (−0.195 + 0.294i)8-s + (2.58 − 0.195i)9-s + (−0.183 − 0.629i)10-s + (0.592 + 0.113i)11-s + (−0.160 − 0.933i)12-s + (0.378 − 1.00i)13-s + (−0.00365 + 0.0212i)14-s + (1.51 + 0.888i)15-s + (−0.241 + 0.0654i)16-s + (0.348 − 1.38i)17-s + ⋯

Functional equation

Λ(s)=(334s/2 ΓC(s) L(s)=((0.910+0.412i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(334s/2 ΓC(s+1/2) L(s)=((0.910+0.412i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 334334    =    2⋅1672 \cdot 167
Sign: 0.910+0.412i0.910 + 0.412i
Analytic conductor: 2.667002.66700
Root analytic conductor: 1.633091.63309
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ334(275,⋅)\chi_{334} (275, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 334, ( :1/2), 0.910+0.412i)(2,\ 334,\ (\ :1/2),\ 0.910 + 0.412i)

Particular Values

L(1)L(1) ≈\approx 0.780195−0.168603i0.780195 - 0.168603i
L(12)L(\frac12) ≈\approx 0.780195−0.168603i0.780195 - 0.168603i
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+(−0.752−0.658i)T 1 + (-0.752 - 0.658i)T
167 1+(12.9+0.719i)T 1 + (12.9 + 0.719i)T
good3 1+(3.27−0.124i)T+(2.99−0.226i)T2 1 + (3.27 - 0.124i)T + (2.99 - 0.226i)T^{2}
5 1+(1.74+1.11i)T+(2.10+4.53i)T2 1 + (1.74 + 1.11i)T + (2.10 + 4.53i)T^{2}
7 1+(−0.0421−0.0689i)T+(−3.18+6.23i)T2 1 + (-0.0421 - 0.0689i)T + (-3.18 + 6.23i)T^{2}
11 1+(−1.96−0.376i)T+(10.2+4.06i)T2 1 + (-1.96 - 0.376i)T + (10.2 + 4.06i)T^{2}
13 1+(−1.36+3.63i)T+(−9.78−8.56i)T2 1 + (-1.36 + 3.63i)T + (-9.78 - 8.56i)T^{2}
17 1+(−1.43+5.72i)T+(−14.9−8.03i)T2 1 + (-1.43 + 5.72i)T + (-14.9 - 8.03i)T^{2}
19 1+(−0.0943−1.65i)T+(−18.8+2.15i)T2 1 + (-0.0943 - 1.65i)T + (-18.8 + 2.15i)T^{2}
23 1+(−5.32+1.88i)T+(17.8−14.4i)T2 1 + (-5.32 + 1.88i)T + (17.8 - 14.4i)T^{2}
29 1+(−5.25−1.86i)T+(22.5+18.2i)T2 1 + (-5.25 - 1.86i)T + (22.5 + 18.2i)T^{2}
31 1+(0.674+0.741i)T+(−2.92+30.8i)T2 1 + (0.674 + 0.741i)T + (-2.92 + 30.8i)T^{2}
37 1+(11.2+0.856i)T+(36.5+5.58i)T2 1 + (11.2 + 0.856i)T + (36.5 + 5.58i)T^{2}
41 1+(0.247−2.60i)T+(−40.2−7.71i)T2 1 + (0.247 - 2.60i)T + (-40.2 - 7.71i)T^{2}
43 1+(0.387−0.459i)T+(−7.28−42.3i)T2 1 + (0.387 - 0.459i)T + (-7.28 - 42.3i)T^{2}
47 1+(−4.08+1.99i)T+(28.9−37.0i)T2 1 + (-4.08 + 1.99i)T + (28.9 - 37.0i)T^{2}
53 1+(−6.17+8.56i)T+(−16.7−50.2i)T2 1 + (-6.17 + 8.56i)T + (-16.7 - 50.2i)T^{2}
59 1+(2.82+11.2i)T+(−52.0+27.8i)T2 1 + (2.82 + 11.2i)T + (-52.0 + 27.8i)T^{2}
61 1+(5.92−1.36i)T+(54.8−26.7i)T2 1 + (5.92 - 1.36i)T + (54.8 - 26.7i)T^{2}
67 1+(−5.43+3.46i)T+(28.2−60.7i)T2 1 + (-5.43 + 3.46i)T + (28.2 - 60.7i)T^{2}
71 1+(−2.03−0.898i)T+(47.7+52.5i)T2 1 + (-2.03 - 0.898i)T + (47.7 + 52.5i)T^{2}
73 1+(−15.2−4.14i)T+(62.9+36.8i)T2 1 + (-15.2 - 4.14i)T + (62.9 + 36.8i)T^{2}
79 1+(9.25−9.43i)T+(−1.49−78.9i)T2 1 + (9.25 - 9.43i)T + (-1.49 - 78.9i)T^{2}
83 1+(8.04−7.04i)T+(10.9−82.2i)T2 1 + (8.04 - 7.04i)T + (10.9 - 82.2i)T^{2}
89 1+(−2.11+1.23i)T+(43.5−77.6i)T2 1 + (-2.11 + 1.23i)T + (43.5 - 77.6i)T^{2}
97 1+(−7.48+8.23i)T+(−9.16−96.5i)T2 1 + (-7.48 + 8.23i)T + (-9.16 - 96.5i)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

−11.71797719084356548170087719200, −10.91116697674650901756367082682, −9.910013297493164090311125117845, −8.484938618595156653321843228768, −7.26145067045419334334531930240, −6.56684158309104848346177960322, −5.35740530596350079791595140034, −4.88163225004317365959592882951, −3.68266030059911567147969238715, −0.70922002108513417616745239382, 1.34385923221794237186496345108, 3.71614050667496382295987113224, 4.55851574115543603601394288084, 5.73108492728255187834607798118, 6.57882699310333475317617082674, 7.31029041550441401517971593803, 9.053838665662984278914361036450, 10.50249285996345166361816730282, 10.81839195373916709991892777395, 11.80958935804312712006777251996

Graph of the ZZ-function along the critical line