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Properties

Label 2-350-35.3-c1-0-0
Degree $2$
Conductor $350$
Sign $-0.0774 - 0.996i$
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 + (−0.788 − 2.94i)3-s + (0.866 + 0.499i)4-s + 3.04i·6-s + (−2.01 + 1.71i)7-s + (−0.707 − 0.707i)8-s + (−5.43 + 3.13i)9-s + (−1.13 + 1.97i)11-s + (0.788 − 2.94i)12-s + (−3.37 + 3.37i)13-s + (2.38 − 1.13i)14-s + (0.500 + 0.866i)16-s + (3.34 − 0.896i)17-s + (6.06 − 1.62i)18-s + (−3.70 − 6.41i)19-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.455 − 1.69i)3-s + (0.433 + 0.249i)4-s + 1.24i·6-s + (−0.760 + 0.648i)7-s + (−0.249 − 0.249i)8-s + (−1.81 + 1.04i)9-s + (−0.342 + 0.594i)11-s + (0.227 − 0.849i)12-s + (−0.936 + 0.936i)13-s + (0.638 − 0.303i)14-s + (0.125 + 0.216i)16-s + (0.811 − 0.217i)17-s + (1.42 − 0.382i)18-s + (−0.849 − 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0774 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0774 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.0774 - 0.996i$
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: \((2,\ 350,\ (\ :1/2),\ -0.0774 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0495571 + 0.0535563i\)
\(L(\frac12)\) \(\approx\) \(0.0495571 + 0.0535563i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.01 - 1.71i)T \)
good3 \( 1 + (0.788 + 2.94i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.37 - 3.37i)T - 13iT^{2} \)
17 \( 1 + (-3.34 + 0.896i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.70 + 6.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.776 - 2.89i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.27iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.19 + 0.588i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (0.512 + 0.512i)T + 43iT^{2} \)
47 \( 1 + (0.123 - 0.459i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (12.3 - 3.31i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.19 + 9i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.08 - 0.627i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.91 + 10.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-0.216 - 0.808i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (8.66 - 5i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.888 - 0.888i)T - 83iT^{2} \)
89 \( 1 + (-4.56 - 7.91i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.18 - 1.18i)T + 97iT^{2} \)
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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

−11.84818230707915768927122211886, −11.06239436895093333533424129905, −9.729398100770392644033665879651, −8.930631330377802550651849536388, −7.75401987963142777348052868590, −7.02021482451649532797000592330, −6.38038252723998590697851503336, −5.12472413307739596223117197413, −2.80475733430882850047103596345, −1.81949803846143164268510012797, 0.06192247636326251883632669373, 3.08452941630768717416033346834, 4.10380501089566645226208454184, 5.43641402132067451637285034853, 6.16576352921103655820196856926, 7.66189997547931125625926116320, 8.640185111558499991857745725820, 9.770296479055789897951183340506, 10.28804456544253431578814988093, 10.63294991194949719934991047927

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