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Properties

Label 2-35-5.2-c4-0-5
Degree $2$
Conductor $35$
Sign $0.876 - 0.480i$
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  + (3.18 + 3.18i)2-s + (5.77 − 5.77i)3-s + 4.30i·4-s + (16.7 + 18.5i)5-s + 36.7·6-s + (−13.0 − 13.0i)7-s + (37.2 − 37.2i)8-s + 14.3i·9-s + (−5.85 + 112. i)10-s − 82.8·11-s + (24.8 + 24.8i)12-s + (−138. + 138. i)13-s − 83.4i·14-s + (203. + 10.6i)15-s + 306.·16-s + (−322. − 322. i)17-s + ⋯
L(s)  = 1  + (0.796 + 0.796i)2-s + (0.641 − 0.641i)3-s + 0.269i·4-s + (0.669 + 0.742i)5-s + 1.02·6-s + (−0.267 − 0.267i)7-s + (0.582 − 0.582i)8-s + 0.177i·9-s + (−0.0585 + 1.12i)10-s − 0.684·11-s + (0.172 + 0.172i)12-s + (−0.820 + 0.820i)13-s − 0.425i·14-s + (0.905 + 0.0471i)15-s + 1.19·16-s + (−1.11 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.876 - 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.876 - 0.480i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.876 - 0.480i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.45469 + 0.628870i\)
\(L(\frac12)\) \(\approx\) \(2.45469 + 0.628870i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-16.7 - 18.5i)T \)
7 \( 1 + (13.0 + 13.0i)T \)
good2 \( 1 + (-3.18 - 3.18i)T + 16iT^{2} \)
3 \( 1 + (-5.77 + 5.77i)T - 81iT^{2} \)
11 \( 1 + 82.8T + 1.46e4T^{2} \)
13 \( 1 + (138. - 138. i)T - 2.85e4iT^{2} \)
17 \( 1 + (322. + 322. i)T + 8.35e4iT^{2} \)
19 \( 1 + 318. iT - 1.30e5T^{2} \)
23 \( 1 + (-138. + 138. i)T - 2.79e5iT^{2} \)
29 \( 1 - 109. iT - 7.07e5T^{2} \)
31 \( 1 + 1.06e3T + 9.23e5T^{2} \)
37 \( 1 + (-1.79e3 - 1.79e3i)T + 1.87e6iT^{2} \)
41 \( 1 + 345.T + 2.82e6T^{2} \)
43 \( 1 + (-1.21e3 + 1.21e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-1.36e3 - 1.36e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (102. - 102. i)T - 7.89e6iT^{2} \)
59 \( 1 + 2.47e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.03e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.42e3 - 3.42e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 5.34e3T + 2.54e7T^{2} \)
73 \( 1 + (-6.03e3 + 6.03e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 9.39e3iT - 3.89e7T^{2} \)
83 \( 1 + (6.84e3 - 6.84e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 2.33e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.86e3 + 3.86e3i)T + 8.85e7iT^{2} \)
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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

−15.55437810695739635557708092449, −14.42777804264319781414328727020, −13.68418307512132763828997407749, −13.01514894861716524764926499306, −10.90881534981920707364052130731, −9.443488665662462962461395063262, −7.39703922853743293126442087912, −6.68817488657458326386577938137, −4.92717878888241663421271704315, −2.49234745545775797056402080060, 2.43217231421583262336528135555, 4.05131330818022506643565919651, 5.53862061414733805697182962735, 8.196422424886671935361723309131, 9.530610504859685295437621245718, 10.69927408222969331115669151673, 12.49546172791323228903890126626, 12.99685095447217087328898605874, 14.33018869902489068764149252591, 15.39496177506347003680576391671

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