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Properties

Label 2-350-35.17-c1-0-2
Degree $2$
Conductor $350$
Sign $0.857 - 0.515i$
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  + (0.258 − 0.965i)2-s + (−2.94 + 0.788i)3-s + (−0.866 − 0.499i)4-s + 3.04i·6-s + (−1.71 − 2.01i)7-s + (−0.707 + 0.707i)8-s + (5.43 − 3.13i)9-s + (−1.13 + 1.97i)11-s + (2.94 + 0.788i)12-s + (3.37 + 3.37i)13-s + (−2.38 + 1.13i)14-s + (0.500 + 0.866i)16-s + (0.896 + 3.34i)17-s + (−1.62 − 6.06i)18-s + (3.70 + 6.41i)19-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−1.69 + 0.455i)3-s + (−0.433 − 0.249i)4-s + 1.24i·6-s + (−0.648 − 0.760i)7-s + (−0.249 + 0.249i)8-s + (1.81 − 1.04i)9-s + (−0.342 + 0.594i)11-s + (0.849 + 0.227i)12-s + (0.936 + 0.936i)13-s + (−0.638 + 0.303i)14-s + (0.125 + 0.216i)16-s + (0.217 + 0.811i)17-s + (−0.382 − 1.42i)18-s + (0.849 + 1.47i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 - 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.857 - 0.515i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.857 - 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.614457 + 0.170445i\)
\(L(\frac12)\) \(\approx\) \(0.614457 + 0.170445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 \)
7 \( 1 + (1.71 + 2.01i)T \)
good3 \( 1 + (2.94 - 0.788i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (1.13 - 1.97i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.37 - 3.37i)T + 13iT^{2} \)
17 \( 1 + (-0.896 - 3.34i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.70 - 6.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.89 - 0.776i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 5.27iT - 29T^{2} \)
31 \( 1 + (3 + 1.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.588 + 2.19i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.19iT - 41T^{2} \)
43 \( 1 + (0.512 - 0.512i)T - 43iT^{2} \)
47 \( 1 + (0.459 + 0.123i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.31 - 12.3i)T + (-45.8 + 26.5i)T^{2} \)
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 \( 1 + (-10.8 + 2.91i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-0.808 + 0.216i)T + (63.2 - 36.5i)T^{2} \)
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

−11.49789245274391296887241869657, −10.69119160096748039601536449837, −10.11396167582512346194363578183, −9.335965715439133054822704569076, −7.58910314452911267580402967663, −6.37034586981302980899210034737, −5.70007503381447769723001544261, −4.42852796185357473363558132540, −3.70933261185606004327290643189, −1.28739932371753801130771945720, 0.60155471473086333610227946179, 3.18895673634807521944275738677, 5.18693440913998762127527310184, 5.42712221785140266000186620683, 6.52028686956362566852690558441, 7.15550756614485727788426334446, 8.464372866084940749250969662868, 9.566073313049987273160417670182, 10.80400070999540814602122415718, 11.41304273603201576068885523453

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