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Properties

Label 2-349-349.123-c1-0-4
Degree $2$
Conductor $349$
Sign $-0.970 - 0.239i$
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  + (−2.10 + 1.21i)2-s + (0.180 − 0.313i)3-s + (1.94 − 3.36i)4-s + (−1.51 + 2.62i)5-s + 0.877i·6-s + (0.636 − 0.367i)7-s + 4.56i·8-s + (1.43 + 2.48i)9-s − 7.34i·10-s − 0.521i·11-s + (−0.702 − 1.21i)12-s + (2.12 − 1.22i)13-s + (−0.891 + 1.54i)14-s + (0.547 + 0.948i)15-s + (−1.65 − 2.86i)16-s − 3.97·17-s + ⋯
L(s)  = 1  + (−1.48 + 0.857i)2-s + (0.104 − 0.180i)3-s + (0.970 − 1.68i)4-s + (−0.677 + 1.17i)5-s + 0.358i·6-s + (0.240 − 0.138i)7-s + 1.61i·8-s + (0.478 + 0.828i)9-s − 2.32i·10-s − 0.157i·11-s + (−0.202 − 0.351i)12-s + (0.588 − 0.339i)13-s + (−0.238 + 0.412i)14-s + (0.141 + 0.245i)15-s + (−0.413 − 0.715i)16-s − 0.963·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.970 - 0.239i$
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: \((2,\ 349,\ (\ :1/2),\ -0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0534869 + 0.439241i\)
\(L(\frac12)\) \(\approx\) \(0.0534869 + 0.439241i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (11.6 - 14.6i)T \)
good2 \( 1 + (2.10 - 1.21i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.180 + 0.313i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.51 - 2.62i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.636 + 0.367i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 0.521iT - 11T^{2} \)
13 \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 3.97T + 17T^{2} \)
19 \( 1 + (1.80 - 3.13i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.71 - 2.97i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.59 - 7.95i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.18T + 31T^{2} \)
37 \( 1 + 0.220T + 37T^{2} \)
41 \( 1 + 6.02T + 41T^{2} \)
43 \( 1 + (3.30 + 1.90i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 - 4.18iT - 53T^{2} \)
59 \( 1 + (-7.00 - 4.04i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 2.04iT - 61T^{2} \)
67 \( 1 - 5.86T + 67T^{2} \)
71 \( 1 + (-5.57 + 3.21i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.09 - 3.62i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.38iT - 79T^{2} \)
83 \( 1 + (-6.22 - 10.7i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.77 - 1.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.50 - 0.866i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.19476112436797943491384560805, −10.86756578637964336469938369248, −10.09437385181730433940746267149, −8.841501797424754572453398746809, −8.094536703947077395812038202645, −7.22335262737416829699040714676, −6.81303303388147574657728594084, −5.46620822607631599487264434582, −3.63965948127751524239444479312, −1.80870692120956164434990402884, 0.49326669777885589200874999512, 1.93533955998440196847033887459, 3.67617293520866445203550365923, 4.71396379362945449358249557604, 6.61045175446064542469105281217, 7.76699895519498919945676805310, 8.721901707400838350329408268476, 9.044101838560885698749385053556, 9.947358418309859637995124432359, 11.16238875347081870835897543460

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