Properties

Label 2-350-35.13-c1-0-2
Degree $2$
Conductor $350$
Sign $0.640 - 0.768i$
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  + (0.707 − 0.707i)2-s + (−1.32 + 1.32i)3-s − 1.00i·4-s + 1.88i·6-s + (1.36 + 2.26i)7-s + (−0.707 − 0.707i)8-s − 0.535i·9-s + 1.73·11-s + (1.32 + 1.32i)12-s + (−3.63 + 3.63i)13-s + (2.56 + 0.633i)14-s − 1.00·16-s + (2.30 + 2.30i)17-s + (−0.378 − 0.378i)18-s + 3.25·19-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.767 + 0.767i)3-s − 0.500i·4-s + 0.767i·6-s + (0.517 + 0.855i)7-s + (−0.250 − 0.250i)8-s − 0.178i·9-s + 0.522·11-s + (0.383 + 0.383i)12-s + (−1.00 + 1.00i)13-s + (0.686 + 0.169i)14-s − 0.250·16-s + (0.558 + 0.558i)17-s + (−0.0893 − 0.0893i)18-s + 0.747·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.640 - 0.768i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.640 - 0.768i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22027 + 0.571448i\)
\(L(\frac12)\) \(\approx\) \(1.22027 + 0.571448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 + (-1.36 - 2.26i)T \)
good3 \( 1 + (1.32 - 1.32i)T - 3iT^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + (3.63 - 3.63i)T - 13iT^{2} \)
17 \( 1 + (-2.30 - 2.30i)T + 17iT^{2} \)
19 \( 1 - 3.25T + 19T^{2} \)
23 \( 1 + (-5.79 - 5.79i)T + 23iT^{2} \)
29 \( 1 - 4.73iT - 29T^{2} \)
31 \( 1 + 8.89iT - 31T^{2} \)
37 \( 1 + (-1.55 + 1.55i)T - 37iT^{2} \)
41 \( 1 - 5.64iT - 41T^{2} \)
43 \( 1 + (2.44 + 2.44i)T + 43iT^{2} \)
47 \( 1 + (6.29 + 6.29i)T + 47iT^{2} \)
53 \( 1 + (10.0 + 10.0i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 6.51iT - 61T^{2} \)
67 \( 1 + (-7.02 + 7.02i)T - 67iT^{2} \)
71 \( 1 - 8.19T + 71T^{2} \)
73 \( 1 + (-7.62 + 7.62i)T - 73iT^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (-3.98 + 3.98i)T - 83iT^{2} \)
89 \( 1 + 5.64T + 89T^{2} \)
97 \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \)
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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

−11.51215708010502603439019002600, −11.07343388473527943239338116266, −9.735007692914236823307477998848, −9.358437979042100946432487935472, −7.85464531111582443357213437794, −6.48416329169891465763998205976, −5.29749715604075918803064947048, −4.86312964463564479848111931220, −3.54396450783336187426042994721, −1.90562284567538727278292147327, 0.951048946975474971975529853595, 3.10145295318276438986256442731, 4.65484154818295144124209241475, 5.46745990138002377353468420054, 6.68594276460573538431174773141, 7.26812732031327559785906219580, 8.124859934780388011629344663751, 9.542447946792740919180206690429, 10.68213633943918633126720812683, 11.59448378494578098140460388792

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