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Properties

Label 2-349-349.123-c1-0-2
Degree $2$
Conductor $349$
Sign $-0.703 + 0.710i$
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  + (−0.327 + 0.189i)2-s + (−1.06 + 1.84i)3-s + (−0.928 + 1.60i)4-s + (−1.07 + 1.86i)5-s − 0.803i·6-s + (−2.19 + 1.26i)7-s − 1.45i·8-s + (−0.758 − 1.31i)9-s − 0.816i·10-s + 1.09i·11-s + (−1.97 − 3.41i)12-s + (5.04 − 2.91i)13-s + (0.479 − 0.830i)14-s + (−2.29 − 3.97i)15-s + (−1.58 − 2.73i)16-s + 0.593·17-s + ⋯
L(s)  = 1  + (−0.231 + 0.133i)2-s + (−0.613 + 1.06i)3-s + (−0.464 + 0.804i)4-s + (−0.482 + 0.835i)5-s − 0.328i·6-s + (−0.829 + 0.478i)7-s − 0.515i·8-s + (−0.252 − 0.438i)9-s − 0.258i·10-s + 0.331i·11-s + (−0.569 − 0.986i)12-s + (1.39 − 0.807i)13-s + (0.128 − 0.221i)14-s + (−0.592 − 1.02i)15-s + (−0.395 − 0.684i)16-s + 0.144·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.703 + 0.710i$
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: \((2,\ 349,\ (\ :1/2),\ -0.703 + 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185572 - 0.444545i\)
\(L(\frac12)\) \(\approx\) \(0.185572 - 0.444545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.327 - 0.189i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.06 - 1.84i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.07 - 1.86i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (2.19 - 1.26i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 1.09iT - 11T^{2} \)
13 \( 1 + (-5.04 + 2.91i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.593T + 17T^{2} \)
19 \( 1 + (0.552 - 0.957i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.80 - 3.12i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.325 + 0.563i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 1.48T + 31T^{2} \)
37 \( 1 + 1.94T + 37T^{2} \)
41 \( 1 + 4.81T + 41T^{2} \)
43 \( 1 + (9.90 + 5.72i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.11iT - 47T^{2} \)
53 \( 1 - 1.09iT - 53T^{2} \)
59 \( 1 + (0.459 + 0.265i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 3.54iT - 61T^{2} \)
67 \( 1 - 9.41T + 67T^{2} \)
71 \( 1 + (7.34 - 4.24i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.47 + 9.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.94iT - 79T^{2} \)
83 \( 1 + (-5.88 - 10.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.27 + 0.734i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.45 + 3.72i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.88257918608566357216199217640, −11.02455417236874948491098791481, −10.24234014881051052228350589751, −9.402627233435420429178114899331, −8.441452261264430926090470429907, −7.37420345611115386366667832352, −6.28051839479990780997962324159, −5.13423174703175974072096660142, −3.75669060394384479375625012544, −3.24912032572949321042058018294, 0.42234052538534932943013947725, 1.47833968320889573982986793588, 3.80757059556792287761729771705, 5.05410509467806398754732586881, 6.26433549133893916276732826315, 6.77570277417119763487609960228, 8.272177064957470160710194338236, 8.940613482513441483789571989431, 10.01885395781602980344281437840, 11.06949218153517315576043215804

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