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Properties

Label 2-350-175.48-c1-0-3
Degree $2$
Conductor $350$
Sign $0.917 + 0.398i$
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  + (0.156 − 0.987i)2-s + (−0.217 − 0.427i)3-s + (−0.951 − 0.309i)4-s + (−0.542 + 2.16i)5-s + (−0.455 + 0.148i)6-s + (0.295 + 2.62i)7-s + (−0.453 + 0.891i)8-s + (1.62 − 2.24i)9-s + (2.05 + 0.875i)10-s + (2.41 − 1.75i)11-s + (0.0749 + 0.473i)12-s + (5.43 − 0.861i)13-s + (2.64 + 0.119i)14-s + (1.04 − 0.240i)15-s + (0.809 + 0.587i)16-s + (−0.545 + 1.07i)17-s + ⋯
L(s)  = 1  + (0.110 − 0.698i)2-s + (−0.125 − 0.246i)3-s + (−0.475 − 0.154i)4-s + (−0.242 + 0.970i)5-s + (−0.186 + 0.0604i)6-s + (0.111 + 0.993i)7-s + (−0.160 + 0.315i)8-s + (0.542 − 0.747i)9-s + (0.650 + 0.276i)10-s + (0.728 − 0.529i)11-s + (0.0216 + 0.136i)12-s + (1.50 − 0.238i)13-s + (0.706 + 0.0318i)14-s + (0.269 − 0.0620i)15-s + (0.202 + 0.146i)16-s + (−0.132 + 0.259i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35874 - 0.282213i\)
\(L(\frac12)\) \(\approx\) \(1.35874 - 0.282213i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.156 + 0.987i)T \)
5 \( 1 + (0.542 - 2.16i)T \)
7 \( 1 + (-0.295 - 2.62i)T \)
good3 \( 1 + (0.217 + 0.427i)T + (-1.76 + 2.42i)T^{2} \)
11 \( 1 + (-2.41 + 1.75i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-5.43 + 0.861i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (0.545 - 1.07i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-1.35 - 4.18i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-6.66 - 1.05i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (6.53 + 2.12i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.79 - 1.55i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.83 + 0.291i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-1.02 + 1.41i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-8.57 - 8.57i)T + 43iT^{2} \)
47 \( 1 + (-1.72 + 0.880i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.640 + 0.326i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (6.62 + 4.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.31 + 3.18i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (6.97 + 3.55i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (0.129 - 0.398i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.46 + 9.26i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-8.47 - 2.75i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (8.69 + 4.42i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (6.48 - 4.70i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (14.6 - 7.48i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.24615420000707945373987678975, −10.95229596773023163945387809863, −9.557126088581344189124843378930, −8.882521959357621312655932192446, −7.69140867036790698115932587290, −6.36973019419463255767608957869, −5.77005310386281161967214019910, −3.91244819730497543423128077526, −3.17681643046048385174090097539, −1.50280393286385982784512125168, 1.23922226492467517103935656112, 3.88682651867934414179027207241, 4.52356200576556595995060358609, 5.54590671438450965153152505039, 6.97680937968314597071997578093, 7.56914290647007029116951749610, 8.836055617211972769999695594452, 9.371396287189936123720466446754, 10.73637916909008944537701755126, 11.39146873627096259340033327967

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