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Properties

Label 2-349-349.348-c1-0-23
Degree $2$
Conductor $349$
Sign $-0.709 + 0.704i$
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  − 2.51i·2-s + 2.41·3-s − 4.34·4-s + 2.06·5-s − 6.08i·6-s − 1.33i·7-s + 5.89i·8-s + 2.83·9-s − 5.20i·10-s − 1.77i·11-s − 10.4·12-s + 4.54i·13-s − 3.37·14-s + 4.99·15-s + 6.16·16-s + 6.51·17-s + ⋯
L(s)  = 1  − 1.78i·2-s + 1.39·3-s − 2.17·4-s + 0.923·5-s − 2.48i·6-s − 0.506i·7-s + 2.08i·8-s + 0.946·9-s − 1.64i·10-s − 0.535i·11-s − 3.02·12-s + 1.25i·13-s − 0.901·14-s + 1.28·15-s + 1.54·16-s + 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 349 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.709 + 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(349\)
Sign: $-0.709 + 0.704i$
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: \((2,\ 349,\ (\ :1/2),\ -0.709 + 0.704i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.772355 - 1.87252i\)
\(L(\frac12)\) \(\approx\) \(0.772355 - 1.87252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad349 \( 1 + (-13.2 + 13.1i)T \)
good2 \( 1 + 2.51iT - 2T^{2} \)
3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 - 2.06T + 5T^{2} \)
7 \( 1 + 1.33iT - 7T^{2} \)
11 \( 1 + 1.77iT - 11T^{2} \)
13 \( 1 - 4.54iT - 13T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 + 7.86T + 19T^{2} \)
23 \( 1 + 3.35T + 23T^{2} \)
29 \( 1 - 1.92T + 29T^{2} \)
31 \( 1 + 0.268T + 31T^{2} \)
37 \( 1 - 4.96T + 37T^{2} \)
41 \( 1 + 2.76T + 41T^{2} \)
43 \( 1 - 12.0iT - 43T^{2} \)
47 \( 1 + 1.04iT - 47T^{2} \)
53 \( 1 - 6.53iT - 53T^{2} \)
59 \( 1 - 6.20iT - 59T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 - 1.52T + 67T^{2} \)
71 \( 1 + 8.05iT - 71T^{2} \)
73 \( 1 - 5.11T + 73T^{2} \)
79 \( 1 + 14.0iT - 79T^{2} \)
83 \( 1 + 4.29T + 83T^{2} \)
89 \( 1 - 3.06iT - 89T^{2} \)
97 \( 1 - 0.353iT - 97T^{2} \)
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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

−11.01125418204191791997539091397, −10.09924637677030330422530541147, −9.543704689998299648288648145740, −8.780202077030517495913862264442, −7.891894686099856216068959710243, −6.16842709770605974005647556722, −4.44847877018711060393585612796, −3.57052091314212794577232328923, −2.48099534094293588841021499833, −1.58601649113635953082782565581, 2.35444917185468319826915986303, 3.89481379882259969687098541658, 5.36616478103732466900929635808, 6.03368106240699484879018944054, 7.25304811901831467566499052848, 8.181443046132960508070712222602, 8.587248625854219776218699743028, 9.658385773812862003048614951856, 10.20005021028819893456201098764, 12.42915857050129511004947588334

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