Properties

Label 2-350-25.21-c1-0-10
Degree $2$
Conductor $350$
Sign $-0.463 + 0.886i$
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  + (0.309 − 0.951i)2-s + (−0.169 − 0.122i)3-s + (−0.809 − 0.587i)4-s + (1.11 − 1.93i)5-s + (−0.169 + 0.122i)6-s + 7-s + (−0.809 + 0.587i)8-s + (−0.913 − 2.81i)9-s + (−1.49 − 1.66i)10-s + (−0.524 + 1.61i)11-s + (0.0646 + 0.198i)12-s + (−0.531 − 1.63i)13-s + (0.309 − 0.951i)14-s + (−0.427 + 0.190i)15-s + (0.309 + 0.951i)16-s + (0.417 − 0.303i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.0976 − 0.0709i)3-s + (−0.404 − 0.293i)4-s + (0.499 − 0.866i)5-s + (−0.0690 + 0.0501i)6-s + 0.377·7-s + (−0.286 + 0.207i)8-s + (−0.304 − 0.937i)9-s + (−0.473 − 0.525i)10-s + (−0.158 + 0.486i)11-s + (0.0186 + 0.0573i)12-s + (−0.147 − 0.453i)13-s + (0.0825 − 0.254i)14-s + (−0.110 + 0.0490i)15-s + (0.0772 + 0.237i)16-s + (0.101 − 0.0735i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.463 + 0.886i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.463 + 0.886i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.740723 - 1.22307i\)
\(L(\frac12)\) \(\approx\) \(0.740723 - 1.22307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (-1.11 + 1.93i)T \)
7 \( 1 - T \)
good3 \( 1 + (0.169 + 0.122i)T + (0.927 + 2.85i)T^{2} \)
11 \( 1 + (0.524 - 1.61i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.531 + 1.63i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-0.417 + 0.303i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.45 + 1.05i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.986 + 3.03i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-0.0180 - 0.0131i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.39 - 3.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.43 + 4.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.76 - 5.43i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 + (-9.53 - 6.92i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.27 - 4.55i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.971 - 2.98i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.49 - 10.7i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-6.72 + 4.88i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-9.06 - 6.58i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.18 + 6.72i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.92 + 4.30i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.48 - 1.07i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.35 - 7.24i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-2.33 - 1.69i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.26302292261274457636416450449, −10.30767259638451642284881434692, −9.328691123310069843131727979905, −8.746527464209497748389875930657, −7.44682555047493770107591201200, −6.01851831221269009850522838245, −5.15910054538890867865874395791, −4.09098305359603607078753694775, −2.57143138777657185526521833651, −1.00662356259089923641693387008, 2.26646435175567736793743997302, 3.71087911852565041993952355186, 5.18744103889982456684869248632, 5.86497591334350206963083299937, 7.06330263776706205339189059092, 7.81079605396015189219953276486, 8.892504807512074981361423060365, 9.975514884040688683837035465584, 10.91007610444186974021222410445, 11.60736033222868548166725608338

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