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Properties

Label 2-35-35.13-c1-0-1
Degree $2$
Conductor $35$
Sign $0.899 - 0.437i$
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 + (−0.581 − 2.58i)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 + (−0.219 − 0.975i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.899 - 0.437i$
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: \((2,\ 35,\ (\ :1/2),\ 0.899 - 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598865 + 0.138099i\)
\(L(\frac12)\) \(\approx\) \(0.598865 + 0.138099i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.581 + 2.58i)T \)
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

−16.76343872893160696850716555329, −15.59295056623720037119251875678, −14.40286705303137546668344750123, −13.33715942497471909432425137471, −11.98832914484567149905474922867, −10.16812235815551413298297690346, −8.524147519473872321912538499874, −7.45845671624323933270033329055, −6.94729152762131401251943129619, −3.37951316042159346516949348695, 3.02738172602077541111222636184, 5.16726999488233885655867866114, 8.175085457484967238510962436060, 9.126247062444299222986819673632, 9.920579216029164867063356025505, 11.45229553346159824489309456196, 12.60739506476101158876973833850, 14.54109240612487208068165637664, 15.36140843477250342964506579862, 16.29056604287506629782510411258

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