Properties

Label 2-350-25.4-c1-0-13
Degree $2$
Conductor $350$
Sign $-0.811 + 0.584i$
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  + (−0.951 − 0.309i)2-s + (1.20 − 1.65i)3-s + (0.809 + 0.587i)4-s + (−1.62 − 1.53i)5-s + (−1.65 + 1.20i)6-s i·7-s + (−0.587 − 0.809i)8-s + (−0.368 − 1.13i)9-s + (1.07 + 1.96i)10-s + (1.14 − 3.52i)11-s + (1.94 − 0.632i)12-s + (−5.17 + 1.68i)13-s + (−0.309 + 0.951i)14-s + (−4.49 + 0.848i)15-s + (0.309 + 0.951i)16-s + (−0.616 − 0.848i)17-s + ⋯
L(s)  = 1  + (−0.672 − 0.218i)2-s + (0.694 − 0.956i)3-s + (0.404 + 0.293i)4-s + (−0.727 − 0.686i)5-s + (−0.676 + 0.491i)6-s − 0.377i·7-s + (−0.207 − 0.286i)8-s + (−0.122 − 0.377i)9-s + (0.339 + 0.620i)10-s + (0.345 − 1.06i)11-s + (0.562 − 0.182i)12-s + (−1.43 + 0.465i)13-s + (−0.0825 + 0.254i)14-s + (−1.16 + 0.218i)15-s + (0.0772 + 0.237i)16-s + (−0.149 − 0.205i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.811 + 0.584i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.281860 - 0.873832i\)
\(L(\frac12)\) \(\approx\) \(0.281860 - 0.873832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 + 0.309i)T \)
5 \( 1 + (1.62 + 1.53i)T \)
7 \( 1 + iT \)
good3 \( 1 + (-1.20 + 1.65i)T + (-0.927 - 2.85i)T^{2} \)
11 \( 1 + (-1.14 + 3.52i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.17 - 1.68i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (0.616 + 0.848i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (-5.60 + 4.07i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (7.79 + 2.53i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-4.23 - 3.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (4.20 - 3.05i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.19 + 0.389i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.91 + 5.88i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 2.01iT - 43T^{2} \)
47 \( 1 + (-7.56 + 10.4i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.15 + 1.58i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.55 + 7.85i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.236 - 0.728i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.23 - 11.3i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (-12.9 - 9.38i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.48 - 0.481i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.52 - 3.28i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.18 - 7.13i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-1.51 + 4.65i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-5.53 + 7.61i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.29824881389625867353642208337, −10.04240153841491800581935187508, −8.967104298410542174874741882559, −8.358922219997178716313435223388, −7.43256222867043897352915310465, −6.89612207458775742183257862631, −5.10944867000664223530136235574, −3.63765187170014563652659259607, −2.28425032093937629651016116026, −0.72460291051482032072390621398, 2.40805139587204035937503940036, 3.62056088618857134995627716755, 4.73713409560773521551424837840, 6.22961239853828447832599887793, 7.58933744296375000663656306345, 7.928252719011303489009515220264, 9.414668838088511721248714173860, 9.779637524701446189744875781510, 10.53533218900241897402428598143, 11.88789332493368022895515368963

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