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Properties

Label 2-310-5.4-c1-0-14
Degree $2$
Conductor $310$
Sign $-0.992 + 0.118i$
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  − i·2-s − 1.78i·3-s − 4-s + (−0.264 − 2.22i)5-s − 1.78·6-s − 1.70i·7-s + i·8-s − 0.174·9-s + (−2.22 + 0.264i)10-s − 0.955·11-s + 1.78i·12-s + 3.25i·13-s − 1.70·14-s + (−3.95 + 0.471i)15-s + 16-s − 0.825i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.02i·3-s − 0.5·4-s + (−0.118 − 0.992i)5-s − 0.727·6-s − 0.643i·7-s + 0.353i·8-s − 0.0580·9-s + (−0.702 + 0.0836i)10-s − 0.288·11-s + 0.514i·12-s + 0.902i·13-s − 0.455·14-s + (−1.02 + 0.121i)15-s + 0.250·16-s − 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(310\)    =    \(2 \cdot 5 \cdot 31\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(2.47536\)
Root analytic conductor: \(1.57332\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{310} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 310,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0668458 - 1.12653i\)
\(L(\frac12)\) \(\approx\) \(0.0668458 - 1.12653i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 + (0.264 + 2.22i)T \)
31 \( 1 - T \)
good3 \( 1 + 1.78iT - 3T^{2} \)
7 \( 1 + 1.70iT - 7T^{2} \)
11 \( 1 + 0.955T + 11T^{2} \)
13 \( 1 - 3.25iT - 13T^{2} \)
17 \( 1 + 0.825iT - 17T^{2} \)
19 \( 1 + 0.877T + 19T^{2} \)
23 \( 1 + 0.580iT - 23T^{2} \)
29 \( 1 - 3.83T + 29T^{2} \)
37 \( 1 + 2.21iT - 37T^{2} \)
41 \( 1 - 3.33T + 41T^{2} \)
43 \( 1 - 7.87iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 + 2.19iT - 53T^{2} \)
59 \( 1 - 0.464T + 59T^{2} \)
61 \( 1 - 2.77T + 61T^{2} \)
67 \( 1 + 14.5iT - 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 12.9iT - 73T^{2} \)
79 \( 1 + 1.56T + 79T^{2} \)
83 \( 1 + 9.44iT - 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 - 10.6iT - 97T^{2} \)
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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

−11.53240124724234424485576391015, −10.35907174616548978450295808991, −9.378911010948735265793816165876, −8.379177967090457726440740527068, −7.50561506889397753966929602314, −6.43196317598894624171780771890, −4.94588483357517826607584749653, −3.96386371826481454815970419483, −2.11765606463432876756980700565, −0.873211372093481929193817551699, 2.84808376295958193657944876939, 4.03163610530703696568759421429, 5.23120826126206387860406263289, 6.17607811391304000459219372095, 7.32935553924737234554635715193, 8.314068662921620801870366634370, 9.388145382730066799375716957758, 10.27200894548884867423352517650, 10.84744850942851096077045853733, 12.10247854312365756521334695418

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