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Properties

Label 2-35-5.3-c4-0-10
Degree $2$
Conductor $35$
Sign $-0.648 + 0.761i$
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  + (3.68 − 3.68i)2-s + (−5.41 − 5.41i)3-s − 11.2i·4-s + (−14.8 − 20.1i)5-s − 39.9·6-s + (13.0 − 13.0i)7-s + (17.5 + 17.5i)8-s − 22.3i·9-s + (−128. − 19.6i)10-s + 58.8·11-s + (−60.8 + 60.8i)12-s + (90.6 + 90.6i)13-s − 96.6i·14-s + (−28.8 + 189. i)15-s + 309.·16-s + (18.7 − 18.7i)17-s + ⋯
L(s)  = 1  + (0.922 − 0.922i)2-s + (−0.601 − 0.601i)3-s − 0.701i·4-s + (−0.592 − 0.805i)5-s − 1.11·6-s + (0.267 − 0.267i)7-s + (0.274 + 0.274i)8-s − 0.275i·9-s + (−1.28 − 0.196i)10-s + 0.486·11-s + (−0.422 + 0.422i)12-s + (0.536 + 0.536i)13-s − 0.493i·14-s + (−0.128 + 0.841i)15-s + 1.20·16-s + (0.0648 − 0.0648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.648 + 0.761i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.648 + 0.761i$
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: \((2,\ 35,\ (\ :2),\ -0.648 + 0.761i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.747653 - 1.61788i\)
\(L(\frac12)\) \(\approx\) \(0.747653 - 1.61788i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (14.8 + 20.1i)T \)
7 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (-3.68 + 3.68i)T - 16iT^{2} \)
3 \( 1 + (5.41 + 5.41i)T + 81iT^{2} \)
11 \( 1 - 58.8T + 1.46e4T^{2} \)
13 \( 1 + (-90.6 - 90.6i)T + 2.85e4iT^{2} \)
17 \( 1 + (-18.7 + 18.7i)T - 8.35e4iT^{2} \)
19 \( 1 + 466. iT - 1.30e5T^{2} \)
23 \( 1 + (-617. - 617. i)T + 2.79e5iT^{2} \)
29 \( 1 - 33.4iT - 7.07e5T^{2} \)
31 \( 1 + 1.02e3T + 9.23e5T^{2} \)
37 \( 1 + (495. - 495. i)T - 1.87e6iT^{2} \)
41 \( 1 - 1.43e3T + 2.82e6T^{2} \)
43 \( 1 + (2.15e3 + 2.15e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (722. - 722. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.10e3 - 2.10e3i)T + 7.89e6iT^{2} \)
59 \( 1 - 3.85e3iT - 1.21e7T^{2} \)
61 \( 1 - 5.49e3T + 1.38e7T^{2} \)
67 \( 1 + (-1.18e3 + 1.18e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.60e3T + 2.54e7T^{2} \)
73 \( 1 + (-6.99e3 - 6.99e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.02e4iT - 3.89e7T^{2} \)
83 \( 1 + (1.42e3 + 1.42e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 5.35e3iT - 6.27e7T^{2} \)
97 \( 1 + (3.09e3 - 3.09e3i)T - 8.85e7iT^{2} \)
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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

−15.13360187070763381496325107763, −13.59961233331333314912871634196, −12.79221127185412491430559715119, −11.71401731060313735294087531912, −11.19970159687087065596444267979, −8.995925367904023075946779828544, −7.17547662924297829187113119377, −5.23298119114836546462805756330, −3.79670728735732138813663508182, −1.21770308427237003909477796347, 3.84299158911120358269660872804, 5.28446369998006865305859594625, 6.55073443154884036084760851697, 8.012800469247288996146062817582, 10.28345435135550411835534004620, 11.28467045941486738553783172046, 12.77588730307236177616743605903, 14.31934719910765946951708570132, 14.98356898928264512309770728842, 16.03925381304076127768303404825

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