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Properties

Label 2-350-35.3-c1-0-5
Degree $2$
Conductor $350$
Sign $0.958 + 0.283i$
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  + (−0.965 − 0.258i)2-s + (0.339 + 1.26i)3-s + (0.866 + 0.499i)4-s − 1.31i·6-s + (−0.884 − 2.49i)7-s + (−0.707 − 0.707i)8-s + (1.10 − 0.637i)9-s + (2.63 − 4.56i)11-s + (−0.339 + 1.26i)12-s + (−0.296 + 0.296i)13-s + (0.209 + 2.63i)14-s + (0.500 + 0.866i)16-s + (3.34 − 0.896i)17-s + (−1.23 + 0.329i)18-s + (2.83 + 4.91i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.196 + 0.732i)3-s + (0.433 + 0.249i)4-s − 0.536i·6-s + (−0.334 − 0.942i)7-s + (−0.249 − 0.249i)8-s + (0.368 − 0.212i)9-s + (0.795 − 1.37i)11-s + (−0.0981 + 0.366i)12-s + (−0.0820 + 0.0820i)13-s + (0.0559 + 0.704i)14-s + (0.125 + 0.216i)16-s + (0.811 − 0.217i)17-s + (−0.290 + 0.0777i)18-s + (0.650 + 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.958 + 0.283i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.958 + 0.283i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08495 - 0.157143i\)
\(L(\frac12)\) \(\approx\) \(1.08495 - 0.157143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.884 + 2.49i)T \)
good3 \( 1 + (-0.339 - 1.26i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-2.63 + 4.56i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.296 - 0.296i)T - 13iT^{2} \)
17 \( 1 + (-3.34 + 0.896i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.83 - 4.91i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.776 - 2.89i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.27iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.09 - 1.36i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (5.85 + 5.85i)T + 43iT^{2} \)
47 \( 1 + (-3.26 + 12.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-9.48 + 2.54i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.19 + 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (12.4 - 7.16i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.964 + 3.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-2.47 - 9.22i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.66 - 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 + (1.97 + 3.41i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-13.5 - 13.5i)T + 97iT^{2} \)
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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

−11.29380095155069987429733721990, −10.18152466869062087719577072826, −9.821248231491245784831748328810, −8.842752961719030733987024728520, −7.81581831950817251209782491858, −6.82356200428129515368816798067, −5.65271793543404100829837497043, −3.92555586008762772933806773347, −3.38309814793028403552163252638, −1.11389925869893923053752047402, 1.53034941847956183385969526463, 2.73861315938511318242733077391, 4.64519276721652797593973951061, 6.00250044706582188968655742979, 7.02831156372704607086971749053, 7.60294067017654766069453666693, 8.815951908646539565895508481402, 9.513583391069482577995453998401, 10.38429073994212411648062539112, 11.70546738858529806331183373127

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