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Properties

Label 2-35-35.13-c1-0-0
Degree $2$
Conductor $35$
Sign $-0.0103 - 0.999i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
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  + (−1 + i)2-s + (−1.58 + 1.58i)3-s + (1.58 − 1.58i)5-s − 3.16i·6-s + (2.58 + 0.581i)7-s + (−2 − 2i)8-s − 2.00i·9-s + 3.16i·10-s − 11-s + (1.58 − 1.58i)13-s + (−3.16 + 2i)14-s + 5.00i·15-s + 4·16-s + (−1.58 − 1.58i)17-s + (2.00 + 2.00i)18-s − 3.16·19-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.912 + 0.912i)3-s + (0.707 − 0.707i)5-s − 1.29i·6-s + (0.975 + 0.219i)7-s + (−0.707 − 0.707i)8-s − 0.666i·9-s + 1.00i·10-s − 0.301·11-s + (0.438 − 0.438i)13-s + (−0.845 + 0.534i)14-s + 1.29i·15-s + 16-s + (−0.383 − 0.383i)17-s + (0.471 + 0.471i)18-s − 0.725·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.0103 - 0.999i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ -0.0103 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342239 + 0.345806i\)
\(L(\frac12)\) \(\approx\) \(0.342239 + 0.345806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.58 + 1.58i)T \)
7 \( 1 + (-2.58 - 0.581i)T \)
good2 \( 1 + (1 - i)T - 2iT^{2} \)
3 \( 1 + (1.58 - 1.58i)T - 3iT^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (-1.58 + 1.58i)T - 13iT^{2} \)
17 \( 1 + (1.58 + 1.58i)T + 17iT^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 + (-2 - 2i)T + 23iT^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + 3.16iT - 31T^{2} \)
37 \( 1 + (6 - 6i)T - 37iT^{2} \)
41 \( 1 - 9.48iT - 41T^{2} \)
43 \( 1 + (3 + 3i)T + 43iT^{2} \)
47 \( 1 + (-4.74 - 4.74i)T + 47iT^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 + 6.32iT - 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 13iT - 79T^{2} \)
83 \( 1 + (-3.16 + 3.16i)T - 83iT^{2} \)
89 \( 1 - 6.32T + 89T^{2} \)
97 \( 1 + (-1.58 - 1.58i)T + 97iT^{2} \)
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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

−17.09459915768791383421083954730, −16.00897177664703438570236490536, −15.16786410824094682503509894907, −13.31589695561115667447681758897, −11.83616758473911985107263360753, −10.50833723239882525662713597838, −9.238250136225366813809977809960, −8.071770588726477605787112555276, −6.04415488762146579249464185117, −4.76830288810563033065637744208, 1.81797194049574087688155293048, 5.61138339060652986428905421835, 6.93564282090983517655828207269, 8.784546719314304664614993136090, 10.58209303738455575164574949203, 11.06694238668489043531651720064, 12.32676045744168026200468324578, 13.81862293297963892515678402907, 14.95842382195979093515524107496, 17.07249777995446349307034207198

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