Properties

Label 2-350-35.13-c1-0-4
Degree $2$
Conductor $350$
Sign $0.999 - 0.0110i$
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 + (−2.28 + 2.28i)3-s − 1.00i·4-s − 3.23i·6-s + (0.835 − 2.51i)7-s + (0.707 + 0.707i)8-s − 7.46i·9-s − 1.73·11-s + (2.28 + 2.28i)12-s + (1.67 − 1.67i)13-s + (1.18 + 2.36i)14-s − 1.00·16-s + (−3.96 − 3.96i)17-s + (5.27 + 5.27i)18-s + 5.60·19-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−1.32 + 1.32i)3-s − 0.500i·4-s − 1.32i·6-s + (0.315 − 0.948i)7-s + (0.250 + 0.250i)8-s − 2.48i·9-s − 0.522·11-s + (0.660 + 0.660i)12-s + (0.464 − 0.464i)13-s + (0.316 + 0.632i)14-s − 0.250·16-s + (−0.960 − 0.960i)17-s + (1.24 + 1.24i)18-s + 1.28·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.999 - 0.0110i$
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: \((2,\ 350,\ (\ :1/2),\ 0.999 - 0.0110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.553556 + 0.00307106i\)
\(L(\frac12)\) \(\approx\) \(0.553556 + 0.00307106i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.835 + 2.51i)T \)
good3 \( 1 + (2.28 - 2.28i)T - 3iT^{2} \)
11 \( 1 + 1.73T + 11T^{2} \)
13 \( 1 + (-1.67 + 1.67i)T - 13iT^{2} \)
17 \( 1 + (3.96 + 3.96i)T + 17iT^{2} \)
19 \( 1 - 5.60T + 19T^{2} \)
23 \( 1 + (-1.55 - 1.55i)T + 23iT^{2} \)
29 \( 1 - 1.26iT - 29T^{2} \)
31 \( 1 - 4.10iT - 31T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 + 9.70iT - 41T^{2} \)
43 \( 1 + (2.44 + 2.44i)T + 43iT^{2} \)
47 \( 1 + (2.90 + 2.90i)T + 47iT^{2} \)
53 \( 1 + (-2.68 - 2.68i)T + 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + (-2.77 + 2.77i)T - 67iT^{2} \)
71 \( 1 + 2.19T + 71T^{2} \)
73 \( 1 + (-5.18 + 5.18i)T - 73iT^{2} \)
79 \( 1 - 2iT - 79T^{2} \)
83 \( 1 + (-6.86 + 6.86i)T - 83iT^{2} \)
89 \( 1 - 9.70T + 89T^{2} \)
97 \( 1 + (3.34 + 3.34i)T + 97iT^{2} \)
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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

−11.04037780777400573579931271135, −10.66703013282457352335250478036, −9.769869131653696758606084247404, −8.994348636573283651240974373846, −7.55534959690094792613541521527, −6.62977467868566855916577841574, −5.41908800991089798559897728933, −4.85130152059748643044506041874, −3.60690258153112071186208516471, −0.59693673742613665601423935523, 1.33394170271769521340681001037, 2.51663416413654839926662124701, 4.72959850320007058533838502127, 5.85199138109300203678683311498, 6.62165334437653331494652307944, 7.77697514971383512454712380932, 8.477141915132868342978840442993, 9.799144250571539259977801182824, 11.05800194095124615747084230100, 11.42749917436028794850910340487

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