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Properties

Label 2-350-35.4-c1-0-0
Degree $2$
Conductor $350$
Sign $-0.652 - 0.758i$
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  + (−0.866 + 0.5i)2-s + (−1.73 − i)3-s + (0.499 − 0.866i)4-s + 1.99·6-s + (−1.73 − 2i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.5 + 2.59i)11-s + (−1.73 + 0.999i)12-s + i·13-s + (2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (5.19 + 3i)17-s + (−0.866 − 0.499i)18-s + (−0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.999 − 0.577i)3-s + (0.249 − 0.433i)4-s + 0.816·6-s + (−0.654 − 0.755i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.452 + 0.783i)11-s + (−0.499 + 0.288i)12-s + 0.277i·13-s + (0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (1.26 + 0.727i)17-s + (−0.204 − 0.117i)18-s + (−0.114 − 0.198i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.652 - 0.758i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.652 - 0.758i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0747494 + 0.162880i\)
\(L(\frac12)\) \(\approx\) \(0.0747494 + 0.162880i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + (1.73 + i)T + (1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (-5.19 - 3i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (7.79 - 4.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.06 - 3.5i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + (-7.79 + 4.5i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-7.79 - 4.5i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.92 + 4i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−11.91386979693574921119668360445, −10.66884644987604937210929857945, −10.15351386429427524738199230285, −9.128886601268688065752206974682, −7.69140614198270563361395868114, −7.15540519822383930588653210999, −6.15333425103588563995511785131, −5.35327274963223102197616660979, −3.70857199231072531366990993898, −1.56897112805393215768802291785, 0.16874270616552545056516906016, 2.55997119724191358317682149515, 3.91466826864249975178918350419, 5.57281077153239740947632813509, 5.92201105205790033241447731572, 7.48935748515932652930798818221, 8.501647577843982655912458937049, 9.586191562849337152813450517363, 10.25302036856817765364477265099, 11.05310341661591466190602755831

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