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Properties

Label 2-930-31.25-c1-0-4
Degree 22
Conductor 930930
Sign 0.275−0.961i0.275 - 0.961i
Analytic cond. 7.426087.42608
Root an. cond. 2.725082.72508
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  + 2-s + (−0.5 − 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s + (0.5 + 0.866i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (−2.5 + 4.33i)11-s + (−0.5 − 0.866i)12-s + (−2 + 3.46i)13-s + (0.5 + 0.866i)14-s + 0.999·15-s + 16-s + (1 + 1.73i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.288 − 0.499i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.753 + 1.30i)11-s + (−0.144 − 0.249i)12-s + (−0.554 + 0.960i)13-s + (0.133 + 0.231i)14-s + 0.258·15-s + 0.250·16-s + (0.242 + 0.420i)17-s + ⋯

Functional equation

Λ(s)=(930s/2 ΓC(s) L(s)=((0.275−0.961i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(930s/2 ΓC(s+1/2) L(s)=((0.275−0.961i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 930930    =    2⋅3⋅5⋅312 \cdot 3 \cdot 5 \cdot 31
Sign: 0.275−0.961i0.275 - 0.961i
Analytic conductor: 7.426087.42608
Root analytic conductor: 2.725082.72508
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ930(211,⋅)\chi_{930} (211, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 930, ( :1/2), 0.275−0.961i)(2,\ 930,\ (\ :1/2),\ 0.275 - 0.961i)

Particular Values

L(1)L(1) ≈\approx 1.38794+1.04654i1.38794 + 1.04654i
L(12)L(\frac12) ≈\approx 1.38794+1.04654i1.38794 + 1.04654i
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−T 1 - T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
31 1+(−3.5+4.33i)T 1 + (-3.5 + 4.33i)T
good7 1+(−0.5−0.866i)T+(−3.5+6.06i)T2 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2}
11 1+(2.5−4.33i)T+(−5.5−9.52i)T2 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2}
13 1+(2−3.46i)T+(−6.5−11.2i)T2 1 + (2 - 3.46i)T + (-6.5 - 11.2i)T^{2}
17 1+(−1−1.73i)T+(−8.5+14.7i)T2 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+(−1−1.73i)T+(−9.5+16.4i)T2 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+4T+23T2 1 + 4T + 23T^{2}
29 1−3T+29T2 1 - 3T + 29T^{2}
37 1+(−2−3.46i)T+(−18.5+32.0i)T2 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2}
41 1+(2−3.46i)T+(−20.5−35.5i)T2 1 + (2 - 3.46i)T + (-20.5 - 35.5i)T^{2}
43 1+(−2−3.46i)T+(−21.5+37.2i)T2 1 + (-2 - 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1−2T+47T2 1 - 2T + 47T^{2}
53 1+(1.5−2.59i)T+(−26.5−45.8i)T2 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2}
59 1+(1.5+2.59i)T+(−29.5+51.0i)T2 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2}
61 1+61T2 1 + 61T^{2}
67 1+(−2+3.46i)T+(−33.5−58.0i)T2 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2}
71 1+(4−6.92i)T+(−35.5−61.4i)T2 1 + (4 - 6.92i)T + (-35.5 - 61.4i)T^{2}
73 1+(1−1.73i)T+(−36.5−63.2i)T2 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2}
79 1+(−39.5+68.4i)T2 1 + (-39.5 + 68.4i)T^{2}
83 1+(4.5−7.79i)T+(−41.5−71.8i)T2 1 + (4.5 - 7.79i)T + (-41.5 - 71.8i)T^{2}
89 1−18T+89T2 1 - 18T + 89T^{2}
97 1+T+97T2 1 + T + 97T^{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

−10.26766778397209428236956838925, −9.679312627270711473262758799626, −8.223942431847154701535969405585, −7.55644136112322467177848497916, −6.77213598447190992149365961721, −5.94985397647227030359309795216, −4.91328561041160248966696011618, −4.15903527634336895777912034697, −2.68985758998949375222145606903, −1.83497885094960504484096930921, 0.65681548417145443507407451125, 2.70545766964029665005559027530, 3.58583723919905213035612411632, 4.70482354311543036645031617494, 5.37593847345605006537868557699, 6.12297997229029298466998079656, 7.40414112668966146621646301903, 8.107670220484163438870640559764, 9.040606803200149543593425769758, 10.27435505761476451009128576905

Graph of the ZZ-function along the critical line