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Properties

Label 2-35-5.3-c4-0-3
Degree $2$
Conductor $35$
Sign $0.00540 - 0.999i$
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  + (−0.0151 + 0.0151i)2-s + (8.40 + 8.40i)3-s + 15.9i·4-s + (−24.2 − 5.87i)5-s − 0.255·6-s + (−13.0 + 13.0i)7-s + (−0.486 − 0.486i)8-s + 60.2i·9-s + (0.458 − 0.280i)10-s + 178.·11-s + (−134. + 134. i)12-s + (130. + 130. i)13-s − 0.398i·14-s + (−154. − 253. i)15-s − 255.·16-s + (219. − 219. i)17-s + ⋯
L(s)  = 1  + (−0.00379 + 0.00379i)2-s + (0.933 + 0.933i)3-s + 0.999i·4-s + (−0.971 − 0.235i)5-s − 0.00709·6-s + (−0.267 + 0.267i)7-s + (−0.00759 − 0.00759i)8-s + 0.743i·9-s + (0.00458 − 0.00280i)10-s + 1.47·11-s + (−0.933 + 0.933i)12-s + (0.770 + 0.770i)13-s − 0.00203i·14-s + (−0.688 − 1.12i)15-s − 0.999·16-s + (0.758 − 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00540 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.00540 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.00540 - 0.999i$
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: \((2,\ 35,\ (\ :2),\ 0.00540 - 0.999i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.16363 + 1.15735i\)
\(L(\frac12)\) \(\approx\) \(1.16363 + 1.15735i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (24.2 + 5.87i)T \)
7 \( 1 + (13.0 - 13.0i)T \)
good2 \( 1 + (0.0151 - 0.0151i)T - 16iT^{2} \)
3 \( 1 + (-8.40 - 8.40i)T + 81iT^{2} \)
11 \( 1 - 178.T + 1.46e4T^{2} \)
13 \( 1 + (-130. - 130. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-219. + 219. i)T - 8.35e4iT^{2} \)
19 \( 1 + 255. iT - 1.30e5T^{2} \)
23 \( 1 + (54.1 + 54.1i)T + 2.79e5iT^{2} \)
29 \( 1 - 438. iT - 7.07e5T^{2} \)
31 \( 1 + 1.44e3T + 9.23e5T^{2} \)
37 \( 1 + (-914. + 914. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.73e3T + 2.82e6T^{2} \)
43 \( 1 + (103. + 103. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.42e3 + 1.42e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-3.30e3 - 3.30e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 4.66e3iT - 1.21e7T^{2} \)
61 \( 1 + 3.97e3T + 1.38e7T^{2} \)
67 \( 1 + (3.72e3 - 3.72e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 1.24e3T + 2.54e7T^{2} \)
73 \( 1 + (3.43e3 + 3.43e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 1.11e4iT - 3.89e7T^{2} \)
83 \( 1 + (-3.63e3 - 3.63e3i)T + 4.74e7iT^{2} \)
89 \( 1 + 8.15e3iT - 6.27e7T^{2} \)
97 \( 1 + (-4.38e3 + 4.38e3i)T - 8.85e7iT^{2} \)
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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

−16.13689510804306654101344410002, −15.02174838700708826112371215710, −13.88547884487333371583165549888, −12.30189062772308898724176410251, −11.35004877395143237350151195950, −9.218848706435614375455531901555, −8.709413239522910118229776176899, −7.13699041056385380801491652352, −4.21378960854958823507130607249, −3.35754000811521578083237134606, 1.28827885874713941833061948301, 3.65350169126799845432439031042, 6.25292833579245983696954897474, 7.59631694588801270959212251108, 8.852800947726236158367508986783, 10.47278694375587432383104282839, 11.90814654722114426109323159513, 13.27110069997723159192348769423, 14.44081853205045681334477934366, 15.03479803294494024171930997724

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