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Properties

Label 2-35-5.3-c4-0-1
Degree $2$
Conductor $35$
Sign $-0.992 + 0.118i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.63 + 3.63i)2-s + (2.99 + 2.99i)3-s − 10.4i·4-s + (−2.80 + 24.8i)5-s − 21.7·6-s + (−13.0 + 13.0i)7-s + (−20.2 − 20.2i)8-s − 63.0i·9-s + (−80.1 − 100. i)10-s − 220.·11-s + (31.2 − 31.2i)12-s + (176. + 176. i)13-s − 95.2i·14-s + (−82.7 + 65.9i)15-s + 314.·16-s + (−101. + 101. i)17-s + ⋯
L(s)  = 1  + (−0.908 + 0.908i)2-s + (0.332 + 0.332i)3-s − 0.652i·4-s + (−0.112 + 0.993i)5-s − 0.604·6-s + (−0.267 + 0.267i)7-s + (−0.316 − 0.316i)8-s − 0.778i·9-s + (−0.801 − 1.00i)10-s − 1.81·11-s + (0.216 − 0.216i)12-s + (1.04 + 1.04i)13-s − 0.485i·14-s + (−0.367 + 0.293i)15-s + 1.22·16-s + (−0.350 + 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.992 + 0.118i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0397355 - 0.665490i\)
\(L(\frac12)\) \(\approx\) \(0.0397355 - 0.665490i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.80 - 24.8i)T \)
7 \( 1 + (13.0 - 13.0i)T \)
good2 \( 1 + (3.63 - 3.63i)T - 16iT^{2} \)
3 \( 1 + (-2.99 - 2.99i)T + 81iT^{2} \)
11 \( 1 + 220.T + 1.46e4T^{2} \)
13 \( 1 + (-176. - 176. i)T + 2.85e4iT^{2} \)
17 \( 1 + (101. - 101. i)T - 8.35e4iT^{2} \)
19 \( 1 - 152. iT - 1.30e5T^{2} \)
23 \( 1 + (-596. - 596. i)T + 2.79e5iT^{2} \)
29 \( 1 - 801. iT - 7.07e5T^{2} \)
31 \( 1 + 175.T + 9.23e5T^{2} \)
37 \( 1 + (-423. + 423. i)T - 1.87e6iT^{2} \)
41 \( 1 - 919.T + 2.82e6T^{2} \)
43 \( 1 + (-628. - 628. i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.20e3 - 2.20e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-554. - 554. i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.16e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.29e3T + 1.38e7T^{2} \)
67 \( 1 + (1.09e3 - 1.09e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.62e3T + 2.54e7T^{2} \)
73 \( 1 + (-1.90e3 - 1.90e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 5.03e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.98e3 + 2.98e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 9.63e3iT - 6.27e7T^{2} \)
97 \( 1 + (2.61e3 - 2.61e3i)T - 8.85e7iT^{2} \)
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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

−16.08060600413248255289702603515, −15.55944561926269500134356030952, −14.55137696906032013370254418594, −12.94778625171777954188250876243, −11.10090813876132708883889746065, −9.790288250840406851716106941914, −8.656992800457354702315212898097, −7.32793394990933405898883331513, −6.10817686054113531203943987321, −3.31033178774821166610036337752, 0.59780056221701444131229207951, 2.59457409144327286896003171097, 5.26260230588955590801244111824, 7.903861513026119529762704793592, 8.672870259802444173356383151390, 10.22091868574611284770524129513, 11.09969680810718636342762869076, 12.85161430072284622625663176742, 13.35648234145423595888753337020, 15.44297400467620523012639796864

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