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Properties

Label 2-350-35.13-c1-0-11
Degree $2$
Conductor $350$
Sign $-0.189 + 0.981i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (0.707 − 0.707i)2-s + (2.28 − 2.28i)3-s − 1.00i·4-s − 3.23i·6-s + (−0.835 + 2.51i)7-s + (−0.707 − 0.707i)8-s − 7.46i·9-s − 1.73·11-s + (−2.28 − 2.28i)12-s + (−1.67 + 1.67i)13-s + (1.18 + 2.36i)14-s − 1.00·16-s + (3.96 + 3.96i)17-s + (−5.27 − 5.27i)18-s + 5.60·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (1.32 − 1.32i)3-s − 0.500i·4-s − 1.32i·6-s + (−0.315 + 0.948i)7-s + (−0.250 − 0.250i)8-s − 2.48i·9-s − 0.522·11-s + (−0.660 − 0.660i)12-s + (−0.464 + 0.464i)13-s + (0.316 + 0.632i)14-s − 0.250·16-s + (0.960 + 0.960i)17-s + (−1.24 − 1.24i)18-s + 1.28·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.52844 - 1.85089i\)
\(L(\frac12)\) \(\approx\) \(1.52844 - 1.85089i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (0.835 - 2.51i)T \)
good3 \( 1 + (-2.28 + 2.28i)T - 3iT^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + (1.67 - 1.67i)T - 13iT^{2} \)
17 \( 1 + (-3.96 - 3.96i)T + 17iT^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 + (1.55 + 1.55i)T + 23iT^{2} \)
29 \( 1 - 1.26iT - 29T^{2} \)
31 \( 1 - 4.10iT - 31T^{2} \)
37 \( 1 + (5.79 - 5.79i)T - 37iT^{2} \)
41 \( 1 + 9.70iT - 41T^{2} \)
43 \( 1 + (-2.44 - 2.44i)T + 43iT^{2} \)
47 \( 1 + (-2.90 - 2.90i)T + 47iT^{2} \)
53 \( 1 + (2.68 + 2.68i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + (2.77 - 2.77i)T - 67iT^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + (5.18 - 5.18i)T - 73iT^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (6.86 - 6.86i)T - 83iT^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + (-3.34 - 3.34i)T + 97iT^{2} \)
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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

−11.75667614439438656530862484181, −10.16400643581885727460612975647, −9.253632501353462023085960903593, −8.421884474166590481401528345319, −7.49998764531933158480127230153, −6.48292894496507707329295791428, −5.38409577087746370130499941291, −3.52918487900869046193247332807, −2.69867649516854955175333186437, −1.60273232780188218491179534559, 2.83116427262599382513963844716, 3.57799127758510945898966359847, 4.63968531301680868794964786003, 5.53064941965101955912780314558, 7.52866785312157680650364004898, 7.68729640609317323402192473058, 9.065298589034335050035143357171, 9.864128760735621153114519006416, 10.43755714077444302505365606447, 11.71008210180357275036412395397

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