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Properties

Label 2-350-175.108-c1-0-11
Degree $2$
Conductor $350$
Sign $0.726 + 0.687i$
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  + (−0.0523 − 0.998i)2-s + (0.430 + 0.348i)3-s + (−0.994 + 0.104i)4-s + (2.13 − 0.656i)5-s + (0.325 − 0.448i)6-s + (−0.343 + 2.62i)7-s + (0.156 + 0.987i)8-s + (−0.559 − 2.63i)9-s + (−0.767 − 2.10i)10-s + (4.51 + 0.959i)11-s + (−0.464 − 0.301i)12-s + (−1.77 − 3.47i)13-s + (2.63 + 0.205i)14-s + (1.14 + 0.462i)15-s + (0.978 − 0.207i)16-s + (7.52 + 2.88i)17-s + ⋯
L(s)  = 1  + (−0.0370 − 0.706i)2-s + (0.248 + 0.201i)3-s + (−0.497 + 0.0522i)4-s + (0.955 − 0.293i)5-s + (0.132 − 0.182i)6-s + (−0.129 + 0.991i)7-s + (0.0553 + 0.349i)8-s + (−0.186 − 0.878i)9-s + (−0.242 − 0.664i)10-s + (1.36 + 0.289i)11-s + (−0.134 − 0.0870i)12-s + (−0.491 − 0.965i)13-s + (0.704 + 0.0548i)14-s + (0.296 + 0.119i)15-s + (0.244 − 0.0519i)16-s + (1.82 + 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49301 - 0.594567i\)
\(L(\frac12)\) \(\approx\) \(1.49301 - 0.594567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0523 + 0.998i)T \)
5 \( 1 + (-2.13 + 0.656i)T \)
7 \( 1 + (0.343 - 2.62i)T \)
good3 \( 1 + (-0.430 - 0.348i)T + (0.623 + 2.93i)T^{2} \)
11 \( 1 + (-4.51 - 0.959i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (1.77 + 3.47i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-7.52 - 2.88i)T + (12.6 + 11.3i)T^{2} \)
19 \( 1 + (0.203 - 1.93i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (5.87 - 0.308i)T + (22.8 - 2.40i)T^{2} \)
29 \( 1 + (2.48 + 3.42i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.81 + 4.08i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (0.822 - 1.26i)T + (-15.0 - 33.8i)T^{2} \)
41 \( 1 + (3.28 + 1.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-6.54 - 6.54i)T + 43iT^{2} \)
47 \( 1 + (0.294 + 0.766i)T + (-34.9 + 31.4i)T^{2} \)
53 \( 1 + (7.12 - 8.79i)T + (-11.0 - 51.8i)T^{2} \)
59 \( 1 + (2.32 + 2.57i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (1.38 + 1.24i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (3.59 - 9.36i)T + (-49.7 - 44.8i)T^{2} \)
71 \( 1 + (-2.57 + 1.87i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (6.54 - 4.25i)T + (29.6 - 66.6i)T^{2} \)
79 \( 1 + (3.15 - 7.09i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (14.5 - 2.31i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (3.16 - 3.51i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (5.61 + 0.888i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.59761415169156160138487601845, −10.00944407598257260351927797628, −9.756107104082823959105495707813, −8.918644257516669936586396839147, −7.933734962795924259742375959949, −6.11208786244167529587797425718, −5.64583094068120863465720873022, −4.03587042036957720794744171017, −2.90379927730270667713351728610, −1.49330699402693616914697273146, 1.61523826911282662026086092630, 3.42550679629079010179243523821, 4.79341112408202053453901081830, 5.91859916135748504339443442480, 6.95402644179543868360973918169, 7.53513298283991619873806789022, 8.844617734407687306871085189233, 9.665928035416991291547415355027, 10.41982248204371547475236158309, 11.57754010322994693506356095547

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