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Properties

Label 2-349-349.123-c1-0-14
Degree $2$
Conductor $349$
Sign $0.954 - 0.297i$
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  + (0.228 − 0.131i)2-s + (0.150 − 0.260i)3-s + (−0.965 + 1.67i)4-s + (0.807 − 1.39i)5-s − 0.0793i·6-s + (1.23 − 0.713i)7-s + 1.03i·8-s + (1.45 + 2.51i)9-s − 0.425i·10-s + 2.29i·11-s + (0.290 + 0.503i)12-s + (0.932 − 0.538i)13-s + (0.187 − 0.325i)14-s + (−0.243 − 0.420i)15-s + (−1.79 − 3.10i)16-s + 6.95·17-s + ⋯
L(s)  = 1  + (0.161 − 0.0931i)2-s + (0.0869 − 0.150i)3-s + (−0.482 + 0.835i)4-s + (0.360 − 0.625i)5-s − 0.0323i·6-s + (0.466 − 0.269i)7-s + 0.366i·8-s + (0.484 + 0.839i)9-s − 0.134i·10-s + 0.691i·11-s + (0.0839 + 0.145i)12-s + (0.258 − 0.149i)13-s + (0.0502 − 0.0869i)14-s + (−0.0627 − 0.108i)15-s + (−0.448 − 0.776i)16-s + 1.68·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $0.954 - 0.297i$
Analytic conductor: \(2.78677\)
Root analytic conductor: \(1.66936\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{349} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 349,\ (\ :1/2),\ 0.954 - 0.297i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53615 + 0.233674i\)
\(L(\frac12)\) \(\approx\) \(1.53615 + 0.233674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-2.39 + 18.5i)T \)
good2 \( 1 + (-0.228 + 0.131i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.150 + 0.260i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.807 + 1.39i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.23 + 0.713i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.29iT - 11T^{2} \)
13 \( 1 + (-0.932 + 0.538i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.95T + 17T^{2} \)
19 \( 1 + (-0.461 + 0.799i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.46 + 4.27i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.87 - 8.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 0.988T + 37T^{2} \)
41 \( 1 + 9.20T + 41T^{2} \)
43 \( 1 + (3.45 + 1.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.91iT - 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (7.66 + 4.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 7.83iT - 61T^{2} \)
67 \( 1 + 15.4T + 67T^{2} \)
71 \( 1 + (0.457 - 0.264i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.909 - 1.57i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.588iT - 79T^{2} \)
83 \( 1 + (-3.82 - 6.62i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (14.4 + 8.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-10.7 + 6.20i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.82477618658466477746065364339, −10.52188384526329469312341238795, −9.715276910145753828210721754193, −8.531315745090998062391789154312, −7.919562532817257426351782025219, −6.99820708376997677739994613072, −5.23202370130203120327352631764, −4.67409732864966866413998239980, −3.31311719084044271162661434765, −1.64899356253387491824796993684, 1.31884373597209950936377019511, 3.22563832913074233271174011710, 4.44139872796699271522510353088, 5.79136688988560669976920344156, 6.25219368243138992034896004823, 7.67639398473735586225123261311, 8.794167983267347868344926956068, 9.895038791153624285193749485648, 10.17335897801267824309856100874, 11.45451917812525561662179882377

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