Properties

Label 2-35-5.3-c4-0-6
Degree $2$
Conductor $35$
Sign $0.998 + 0.0606i$
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  + (−5.44 + 5.44i)2-s + (1.36 + 1.36i)3-s − 43.3i·4-s + (7.20 − 23.9i)5-s − 14.8·6-s + (−13.0 + 13.0i)7-s + (149. + 149. i)8-s − 77.2i·9-s + (91.1 + 169. i)10-s + 81.6·11-s + (59.2 − 59.2i)12-s + (−66.7 − 66.7i)13-s − 142. i·14-s + (42.5 − 22.8i)15-s − 929.·16-s + (347. − 347. i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (0.151 + 0.151i)3-s − 2.70i·4-s + (0.288 − 0.957i)5-s − 0.413·6-s + (−0.267 + 0.267i)7-s + (2.32 + 2.32i)8-s − 0.953i·9-s + (0.911 + 1.69i)10-s + 0.674·11-s + (0.411 − 0.411i)12-s + (−0.394 − 0.394i)13-s − 0.727i·14-s + (0.189 − 0.101i)15-s − 3.63·16-s + (1.20 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.998 + 0.0606i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.998 + 0.0606i$
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: \((2,\ 35,\ (\ :2),\ 0.998 + 0.0606i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.712832 - 0.0216219i\)
\(L(\frac12)\) \(\approx\) \(0.712832 - 0.0216219i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-7.20 + 23.9i)T \)
7 \( 1 + (13.0 - 13.0i)T \)
good2 \( 1 + (5.44 - 5.44i)T - 16iT^{2} \)
3 \( 1 + (-1.36 - 1.36i)T + 81iT^{2} \)
11 \( 1 - 81.6T + 1.46e4T^{2} \)
13 \( 1 + (66.7 + 66.7i)T + 2.85e4iT^{2} \)
17 \( 1 + (-347. + 347. i)T - 8.35e4iT^{2} \)
19 \( 1 + 69.2iT - 1.30e5T^{2} \)
23 \( 1 + (-40.2 - 40.2i)T + 2.79e5iT^{2} \)
29 \( 1 + 586. iT - 7.07e5T^{2} \)
31 \( 1 - 538.T + 9.23e5T^{2} \)
37 \( 1 + (1.08e3 - 1.08e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 80.0T + 2.82e6T^{2} \)
43 \( 1 + (1.32e3 + 1.32e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (2.67e3 - 2.67e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-1.86e3 - 1.86e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 1.13e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.39e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.84e3 + 1.84e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 6.93e3T + 2.54e7T^{2} \)
73 \( 1 + (823. + 823. i)T + 2.83e7iT^{2} \)
79 \( 1 + 86.0iT - 3.89e7T^{2} \)
83 \( 1 + (-7.10e3 - 7.10e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.21e4iT - 6.27e7T^{2} \)
97 \( 1 + (7.96e3 - 7.96e3i)T - 8.85e7iT^{2} \)
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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

−16.00509243605440264800701972457, −15.05890587634365281481082686338, −13.91900972923811715004392376207, −11.98843398778409494530802538269, −9.837097542459196234488318421279, −9.314454186180193119546226084275, −8.126109513825561540817987589709, −6.59670579371973395478164001049, −5.25403100225276206955516758634, −0.821076039813503181004616344771, 1.84491461767771514370982471101, 3.46911677310301837865191265664, 7.06428180915901076193905466893, 8.309595147520267219181872086347, 9.855514672275568220699154870951, 10.56166925910822548342054412359, 11.71828803851884540961558800786, 12.99467408739213994174957367795, 14.35643168635537314753148429220, 16.47762434602355615438468556673

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