Properties

Label 2-350-25.21-c1-0-2
Degree $2$
Conductor $350$
Sign $-0.912 - 0.408i$
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 + (1.71 + 1.24i)3-s + (−0.809 − 0.587i)4-s + (−2.14 + 0.644i)5-s + (−1.71 + 1.24i)6-s − 7-s + (0.809 − 0.587i)8-s + (0.469 + 1.44i)9-s + (0.0490 − 2.23i)10-s + (−1.71 + 5.26i)11-s + (−0.656 − 2.02i)12-s + (0.963 + 2.96i)13-s + (0.309 − 0.951i)14-s + (−4.48 − 1.56i)15-s + (0.309 + 0.951i)16-s + (−2.23 + 1.62i)17-s + ⋯
L(s)  = 1  + (−0.218 + 0.672i)2-s + (0.993 + 0.721i)3-s + (−0.404 − 0.293i)4-s + (−0.957 + 0.288i)5-s + (−0.702 + 0.510i)6-s − 0.377·7-s + (0.286 − 0.207i)8-s + (0.156 + 0.481i)9-s + (0.0155 − 0.706i)10-s + (−0.515 + 1.58i)11-s + (−0.189 − 0.583i)12-s + (0.267 + 0.822i)13-s + (0.0825 − 0.254i)14-s + (−1.15 − 0.404i)15-s + (0.0772 + 0.237i)16-s + (−0.542 + 0.393i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.912 - 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.912 - 0.408i$
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: \((2,\ 350,\ (\ :1/2),\ -0.912 - 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.227334 + 1.06429i\)
\(L(\frac12)\) \(\approx\) \(0.227334 + 1.06429i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.14 - 0.644i)T \)
7 \( 1 + T \)
good3 \( 1 + (-1.71 - 1.24i)T + (0.927 + 2.85i)T^{2} \)
11 \( 1 + (1.71 - 5.26i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.963 - 2.96i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.23 - 1.62i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (3.56 - 2.59i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-0.681 + 2.09i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.93 - 5.04i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-8.38 + 6.09i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.75 + 5.39i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.729 - 2.24i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.41T + 43T^{2} \)
47 \( 1 + (5.97 + 4.34i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-7.58 - 5.50i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.42 - 7.46i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.775 - 2.38i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (5.12 - 3.72i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.75 + 2.00i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.452 - 1.39i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.51 - 5.46i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.1 - 8.86i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-0.851 + 2.62i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-7.01 - 5.09i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.96103658521446275503084468390, −10.54444408995766616092077485822, −9.954415251563315108348067558016, −8.896108304657995636599495500457, −8.281595603499062525990453768806, −7.25565317724194244632899618157, −6.40799209136845909897390623065, −4.52430794428126764327779170625, −4.07371075085979816518698926558, −2.54926066185725510067979645716, 0.73172569485840210894964311647, 2.73440251880966496197598693014, 3.35446769426285471129075146413, 4.83504772133537546060676145873, 6.46871995142878124603895035039, 7.73386336650514416263962335952, 8.445151979393639893717591844197, 8.797475518850386241583896629317, 10.29769620041320687768049311573, 11.14702991263624777083468081815

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