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Properties

Label 2-350-25.6-c1-0-5
Degree $2$
Conductor $350$
Sign $0.682 - 0.731i$
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 + (0.408 − 0.296i)3-s + (−0.809 + 0.587i)4-s + (−0.206 + 2.22i)5-s + (−0.408 − 0.296i)6-s − 7-s + (0.809 + 0.587i)8-s + (−0.848 + 2.61i)9-s + (2.18 − 0.491i)10-s + (0.608 + 1.87i)11-s + (−0.156 + 0.480i)12-s + (−0.779 + 2.40i)13-s + (0.309 + 0.951i)14-s + (0.576 + 0.971i)15-s + (0.309 − 0.951i)16-s + (−2.55 − 1.85i)17-s + ⋯
L(s)  = 1  + (−0.218 − 0.672i)2-s + (0.235 − 0.171i)3-s + (−0.404 + 0.293i)4-s + (−0.0924 + 0.995i)5-s + (−0.166 − 0.121i)6-s − 0.377·7-s + (0.286 + 0.207i)8-s + (−0.282 + 0.870i)9-s + (0.689 − 0.155i)10-s + (0.183 + 0.565i)11-s + (−0.0450 + 0.138i)12-s + (−0.216 + 0.665i)13-s + (0.0825 + 0.254i)14-s + (0.148 + 0.250i)15-s + (0.0772 − 0.237i)16-s + (−0.619 − 0.450i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.682 - 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.682 - 0.731i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (281, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.682 - 0.731i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930413 + 0.404422i\)
\(L(\frac12)\) \(\approx\) \(0.930413 + 0.404422i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.206 - 2.22i)T \)
7 \( 1 + T \)
good3 \( 1 + (-0.408 + 0.296i)T + (0.927 - 2.85i)T^{2} \)
11 \( 1 + (-0.608 - 1.87i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.779 - 2.40i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.55 + 1.85i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-5.45 - 3.95i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.753 - 2.31i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.36 + 0.993i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.32 + 0.964i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.66 + 8.20i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.775 - 2.38i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 8.20T + 43T^{2} \)
47 \( 1 + (-0.413 + 0.300i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.70 + 1.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (3.41 - 10.5i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (0.274 + 0.843i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.5 + 7.69i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (5.71 - 4.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.244 - 0.752i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.34 + 3.16i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.62 + 2.63i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.29 - 13.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-6.64 + 4.82i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.50658861190393121532640987577, −10.74120426495699422878388567023, −9.833094742688487199953563883500, −9.079150862830935220133528017198, −7.69556071532479944585858130955, −7.16548242295485782374701792299, −5.76366191477294306401143218221, −4.30891925036724113388071155040, −3.05966497519910207359228993507, −2.02904665657229155899156515999, 0.75364913466258876825673635147, 3.16511770183992870226450089474, 4.46957453228570317255521520515, 5.56987089532088702169513294587, 6.50824910158644646697575962518, 7.70595475239075169203113580745, 8.752616221979218884287868403499, 9.163785607744085613667795448785, 10.14559429343169332536034648437, 11.42719045296105814845150010411

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