Properties

Label 2-349-349.348-c1-0-12
Degree $2$
Conductor $349$
Sign $0.352 - 0.935i$
Analytic cond. $2.78677$
Root an. cond. $1.66936$
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  + 1.25i·2-s + 0.849·3-s + 0.421·4-s + 3.08·5-s + 1.06i·6-s + 1.10i·7-s + 3.04i·8-s − 2.27·9-s + 3.88i·10-s − 3.82i·11-s + 0.357·12-s − 1.93i·13-s − 1.39·14-s + 2.62·15-s − 2.98·16-s − 1.58·17-s + ⋯
L(s)  = 1  + 0.888i·2-s + 0.490·3-s + 0.210·4-s + 1.38·5-s + 0.435i·6-s + 0.418i·7-s + 1.07i·8-s − 0.759·9-s + 1.22i·10-s − 1.15i·11-s + 0.103·12-s − 0.538i·13-s − 0.371·14-s + 0.677·15-s − 0.745·16-s − 0.385·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.352 - 0.935i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (348, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 0.352 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.64445 + 1.13793i\)
\(L(\frac12)\) \(\approx\) \(1.64445 + 1.13793i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (6.58 - 17.4i)T \)
good2 \( 1 - 1.25iT - 2T^{2} \)
3 \( 1 - 0.849T + 3T^{2} \)
5 \( 1 - 3.08T + 5T^{2} \)
7 \( 1 - 1.10iT - 7T^{2} \)
11 \( 1 + 3.82iT - 11T^{2} \)
13 \( 1 + 1.93iT - 13T^{2} \)
17 \( 1 + 1.58T + 17T^{2} \)
19 \( 1 - 0.716T + 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + 6.48T + 29T^{2} \)
31 \( 1 + 3.13T + 31T^{2} \)
37 \( 1 + 0.373T + 37T^{2} \)
41 \( 1 + 3.26T + 41T^{2} \)
43 \( 1 - 1.06iT - 43T^{2} \)
47 \( 1 - 6.35iT - 47T^{2} \)
53 \( 1 + 5.86iT - 53T^{2} \)
59 \( 1 - 3.60iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 - 7.96iT - 71T^{2} \)
73 \( 1 - 6.19T + 73T^{2} \)
79 \( 1 + 2.15iT - 79T^{2} \)
83 \( 1 + 5.06T + 83T^{2} \)
89 \( 1 - 3.12iT - 89T^{2} \)
97 \( 1 + 14.4iT - 97T^{2} \)
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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

−11.45056820172270263648009176452, −10.82107461942977508337483312728, −9.495593353434955394692137275018, −8.755937266209799720224865500599, −7.994927659138729888683015941520, −6.72518604598342646661411333030, −5.73098363531526538208451327882, −5.44203327919278504664093347191, −3.13569352360128072570381479597, −2.11759444264083283151424150789, 1.76065131847874187821438951640, 2.50880980467595152328061360618, 3.86374032048406264786894356614, 5.35774690099437180126179162785, 6.55245798108267243111929339320, 7.41283277927009784888674727766, 9.005139533882282845765953711099, 9.572114510827495735764337017486, 10.37972569341837571052764586295, 11.21293343888140991166489330312

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