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Properties

Label 2-318-53.52-c1-0-3
Degree $2$
Conductor $318$
Sign $0.961 + 0.274i$
Analytic cond. $2.53924$
Root an. cond. $1.59350$
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 − i·3-s − 4-s + 4i·5-s − 6-s + 7-s + i·8-s − 9-s + 4·10-s + 5·11-s + i·12-s + 2·13-s − i·14-s + 4·15-s + 16-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.78i·5-s − 0.408·6-s + 0.377·7-s + 0.353i·8-s − 0.333·9-s + 1.26·10-s + 1.50·11-s + 0.288i·12-s + 0.554·13-s − 0.267i·14-s + 1.03·15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 318 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 318 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(318\)    =    \(2 \cdot 3 \cdot 53\)
Sign: $0.961 + 0.274i$
Analytic conductor: \(2.53924\)
Root analytic conductor: \(1.59350\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{318} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 318,\ (\ :1/2),\ 0.961 + 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34186 - 0.187934i\)
\(L(\frac12)\) \(\approx\) \(1.34186 - 0.187934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
53 \( 1 + (-7 - 2i)T \)
good5 \( 1 - 4iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - iT - 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 - 3iT - 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 + 13iT - 61T^{2} \)
67 \( 1 - 13iT - 67T^{2} \)
71 \( 1 - 7iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 - 5T + 97T^{2} \)
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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

−11.51950611919021271440673056792, −10.90281845878098100411503523348, −9.976678161283512843816827361792, −8.890177834676785330253663348662, −7.68077436283308438393520759525, −6.74422658776545809366699068131, −5.93426590030487536300365765583, −4.02812088271024247271571331199, −3.04304548084901527146420934052, −1.69717907455138203769571317989, 1.20947970709776881780644882711, 3.90878468206067804936956687278, 4.68040532212025185086542620358, 5.57152941841203308210347575606, 6.73866635300919625217018803122, 8.239704885576410988410465565015, 8.841471073307060467201279866998, 9.355928540933491740730661229417, 10.64853138503910323890143443976, 11.97832324529025757152791005566

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