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Properties

Label 2-310-155.109-c1-0-11
Degree $2$
Conductor $310$
Sign $0.137 + 0.990i$
Analytic cond. $2.47536$
Root an. cond. $1.57332$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.587 − 0.809i)2-s + (1.95 − 2.68i)3-s + (−0.309 + 0.951i)4-s + (0.620 + 2.14i)5-s − 3.32·6-s + (4.06 + 1.32i)7-s + (0.951 − 0.309i)8-s + (−2.48 − 7.64i)9-s + (1.37 − 1.76i)10-s + (−0.702 + 2.16i)11-s + (1.95 + 2.68i)12-s + (1.76 − 2.42i)13-s + (−1.32 − 4.06i)14-s + (6.98 + 2.52i)15-s + (−0.809 − 0.587i)16-s + (−2.42 + 0.789i)17-s + ⋯
L(s)  = 1  + (−0.415 − 0.572i)2-s + (1.12 − 1.55i)3-s + (−0.154 + 0.475i)4-s + (0.277 + 0.960i)5-s − 1.35·6-s + (1.53 + 0.499i)7-s + (0.336 − 0.109i)8-s + (−0.827 − 2.54i)9-s + (0.434 − 0.558i)10-s + (−0.211 + 0.652i)11-s + (0.563 + 0.775i)12-s + (0.489 − 0.673i)13-s + (−0.353 − 1.08i)14-s + (1.80 + 0.652i)15-s + (−0.202 − 0.146i)16-s + (−0.589 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $0.137 + 0.990i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ 0.137 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26554 - 1.10231i\)
\(L(\frac12)\) \(\approx\) \(1.26554 - 1.10231i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.587 + 0.809i)T \)
5 \( 1 + (-0.620 - 2.14i)T \)
31 \( 1 + (4.95 + 2.53i)T \)
good3 \( 1 + (-1.95 + 2.68i)T + (-0.927 - 2.85i)T^{2} \)
7 \( 1 + (-4.06 - 1.32i)T + (5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.702 - 2.16i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.76 + 2.42i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.42 - 0.789i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (2.34 - 1.70i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.700 - 0.227i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (7.95 - 5.77i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 4.25iT - 37T^{2} \)
41 \( 1 + (-2.70 + 1.96i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.615 - 0.847i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.225 + 0.309i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-5.87 + 1.90i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (5.33 + 3.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 5.63T + 61T^{2} \)
67 \( 1 + 7.85iT - 67T^{2} \)
71 \( 1 + (-3.63 - 11.1i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.02 - 0.331i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.40 + 7.39i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.94 - 4.05i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (1.64 - 5.05i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.51 + 1.79i)T + (78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.44613300032874002926369592090, −10.76871871335837021900796775907, −9.339043600411971646348043466519, −8.483297851094447110831550561214, −7.74186922105595956988819248853, −7.08989123098917070148707663704, −5.74654597862085349003949133569, −3.66849621478661343149028945285, −2.29628897031919199578043201896, −1.76774488120649024146637732254, 1.98447076180743476132720616835, 4.04138887057570670194062069998, 4.67374737586247197388953057661, 5.61356438882280028526120297283, 7.61758526973355500896752452733, 8.464076212140230871585679951952, 8.865965788098362341857088503557, 9.724996993043408597580550953373, 10.82563198658503707625616137994, 11.33874265937607418790684644046

Graph of the $Z$-function along the critical line