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Properties

Label 2-35-5.2-c4-0-4
Degree $2$
Conductor $35$
Sign $-0.555 - 0.831i$
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  + (5.01 + 5.01i)2-s + (−5.80 + 5.80i)3-s + 34.2i·4-s + (17.0 − 18.2i)5-s − 58.1·6-s + (−13.0 − 13.0i)7-s + (−91.7 + 91.7i)8-s + 13.7i·9-s + (177. − 6.26i)10-s + 75.9·11-s + (−198. − 198. i)12-s + (87.6 − 87.6i)13-s − 131. i·14-s + (7.24 + 204. i)15-s − 371.·16-s + (230. + 230. i)17-s + ⋯
L(s)  = 1  + (1.25 + 1.25i)2-s + (−0.644 + 0.644i)3-s + 2.14i·4-s + (0.681 − 0.731i)5-s − 1.61·6-s + (−0.267 − 0.267i)7-s + (−1.43 + 1.43i)8-s + 0.169i·9-s + (1.77 − 0.0626i)10-s + 0.627·11-s + (−1.38 − 1.38i)12-s + (0.518 − 0.518i)13-s − 0.670i·14-s + (0.0322 + 0.910i)15-s − 1.45·16-s + (0.798 + 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.555 - 0.831i$
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: \((2,\ 35,\ (\ :2),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.09127 + 2.04134i\)
\(L(\frac12)\) \(\approx\) \(1.09127 + 2.04134i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-17.0 + 18.2i)T \)
7 \( 1 + (13.0 + 13.0i)T \)
good2 \( 1 + (-5.01 - 5.01i)T + 16iT^{2} \)
3 \( 1 + (5.80 - 5.80i)T - 81iT^{2} \)
11 \( 1 - 75.9T + 1.46e4T^{2} \)
13 \( 1 + (-87.6 + 87.6i)T - 2.85e4iT^{2} \)
17 \( 1 + (-230. - 230. i)T + 8.35e4iT^{2} \)
19 \( 1 + 527. iT - 1.30e5T^{2} \)
23 \( 1 + (587. - 587. i)T - 2.79e5iT^{2} \)
29 \( 1 + 1.58e3iT - 7.07e5T^{2} \)
31 \( 1 + 588.T + 9.23e5T^{2} \)
37 \( 1 + (-629. - 629. i)T + 1.87e6iT^{2} \)
41 \( 1 + 235.T + 2.82e6T^{2} \)
43 \( 1 + (1.36e3 - 1.36e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (1.62e3 + 1.62e3i)T + 4.87e6iT^{2} \)
53 \( 1 + (-845. + 845. i)T - 7.89e6iT^{2} \)
59 \( 1 - 1.60e3iT - 1.21e7T^{2} \)
61 \( 1 + 2.51e3T + 1.38e7T^{2} \)
67 \( 1 + (-3.67e3 - 3.67e3i)T + 2.01e7iT^{2} \)
71 \( 1 - 6.83e3T + 2.54e7T^{2} \)
73 \( 1 + (1.60e3 - 1.60e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 3.82e3iT - 3.89e7T^{2} \)
83 \( 1 + (4.12e3 - 4.12e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 1.14e3iT - 6.27e7T^{2} \)
97 \( 1 + (-1.53e3 - 1.53e3i)T + 8.85e7iT^{2} \)
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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

−16.11077528535057983146928947811, −15.19733092089510572578113851613, −13.74307414689246097966366920298, −13.09540401871052491246968546100, −11.66360001433491646699162566885, −9.850609114303738196464175011788, −8.059498533879821485955600267548, −6.24435369341738183627820175351, −5.33615927386294033962843675484, −4.05360330863708751511159387863, 1.63437600263852094968414398173, 3.49843335734327967575680150334, 5.64506667882688109038320926463, 6.58885297570836202420086804332, 9.635532517806233924147922229432, 10.84943131258365771798244367711, 11.97653378117521647088702061227, 12.64277590303664877419201702015, 14.04667261534211621660723570701, 14.58766716268095234413807279864

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