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Properties

Label 2-338-13.12-c1-0-4
Degree $2$
Conductor $338$
Sign $0.554 + 0.832i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
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 − 3·3-s − 4-s + i·5-s + 3i·6-s + i·7-s + i·8-s + 6·9-s + 10-s − 2i·11-s + 3·12-s + 14-s − 3i·15-s + 16-s + 3·17-s − 6i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.73·3-s − 0.5·4-s + 0.447i·5-s + 1.22i·6-s + 0.377i·7-s + 0.353i·8-s + 2·9-s + 0.316·10-s − 0.603i·11-s + 0.866·12-s + 0.267·14-s − 0.774i·15-s + 0.250·16-s + 0.727·17-s − 1.41i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.554 + 0.832i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.554 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623881 - 0.333891i\)
\(L(\frac12)\) \(\approx\) \(0.623881 - 0.333891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
13 \( 1 \)
good3 \( 1 + 3T + 3T^{2} \)
5 \( 1 - iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 3iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 - 13iT - 47T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + 10iT - 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 5iT - 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−11.18047868497216008007106159463, −10.93024172789008487501589186650, −9.924146264875952087031334182411, −8.870509198286320455399024523957, −7.34720825932023129995424141034, −6.32464915777190875153043321239, −5.42935988301140703717599064676, −4.50690536697233913166806316122, −2.89432907762669815444107052238, −0.859577517182058029245870736867, 1.06052490666607269611678605569, 4.03160074187732336827544777163, 5.07291643091549864594806785953, 5.71925884521297657304801242013, 6.79707748033097580981175146742, 7.51890132567850397173046782470, 8.860129002414770466827237074886, 10.14551386766293799694674833605, 10.58675556381833170790613124528, 11.93331214684303278908582218698

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