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Properties

Label 2-350-25.21-c1-0-5
Degree $2$
Conductor $350$
Sign $0.999 - 0.0418i$
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 + (1.47 + 1.07i)3-s + (−0.809 − 0.587i)4-s + (1.11 + 1.93i)5-s + (1.47 − 1.07i)6-s + 7-s + (−0.809 + 0.587i)8-s + (0.104 + 0.321i)9-s + (2.18 − 0.464i)10-s + (−0.284 + 0.875i)11-s + (−0.564 − 1.73i)12-s + (0.486 + 1.49i)13-s + (0.309 − 0.951i)14-s + (−0.427 + 4.06i)15-s + (0.309 + 0.951i)16-s + (3.08 − 2.23i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (0.853 + 0.620i)3-s + (−0.404 − 0.293i)4-s + (0.499 + 0.866i)5-s + (0.603 − 0.438i)6-s + 0.377·7-s + (−0.286 + 0.207i)8-s + (0.0348 + 0.107i)9-s + (0.691 − 0.147i)10-s + (−0.0857 + 0.263i)11-s + (−0.162 − 0.501i)12-s + (0.134 + 0.415i)13-s + (0.0825 − 0.254i)14-s + (−0.110 + 1.04i)15-s + (0.0772 + 0.237i)16-s + (0.747 − 0.543i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0418i$
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: \((2,\ 350,\ (\ :1/2),\ 0.999 - 0.0418i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01487 + 0.0422055i\)
\(L(\frac12)\) \(\approx\) \(2.01487 + 0.0422055i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 1.07i)T + (0.927 + 2.85i)T^{2} \)
11 \( 1 + (0.284 - 0.875i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.486 - 1.49i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.08 + 2.23i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.83 - 1.33i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.0315 - 0.0971i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.01 + 1.46i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.20 + 2.33i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.656 + 2.01i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.69 + 8.29i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.79T + 43T^{2} \)
47 \( 1 + (5.68 + 4.12i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.315 - 0.228i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.101 - 0.313i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.09 - 6.43i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.132 + 0.0962i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (3.87 + 2.81i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.83 + 11.7i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.66 - 6.29i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.5 - 9.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.07 - 9.46i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-7.66 - 5.56i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.41908826817140627541742863984, −10.39895155877188630645988691422, −9.810823747015664320296796713545, −9.005964828225553402367119990550, −7.932333046046691905139683522755, −6.65322217801187417690850385195, −5.40133005811194615877789120880, −4.08284927434453089181097596709, −3.14687240495120254872102035976, −2.04477253170785379228223072833, 1.55580622744009536859844729727, 3.12658237187100748757173901353, 4.68167955671580556883965320169, 5.61721076352969658613593299576, 6.73695244024724284193001340495, 8.108175388595447669952600858179, 8.231873394896592070885573541770, 9.286357680945438552398626296766, 10.37688992749001284619416568320, 11.74440369767403066607830861914

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