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Properties

Label 2-35-35.27-c1-0-1
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: $\chi_{35} (27, \cdot )$
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

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

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