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Properties

Label 2-35-7.5-c4-0-7
Degree $2$
Conductor $35$
Sign $-0.481 + 0.876i$
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  + (3.33 − 5.77i)2-s + (2.41 − 1.39i)3-s + (−14.2 − 24.5i)4-s + (9.68 + 5.59i)5-s − 18.5i·6-s + (−7.25 − 48.4i)7-s − 82.6·8-s + (−36.6 + 63.4i)9-s + (64.5 − 37.2i)10-s + (100. + 174. i)11-s + (−68.4 − 39.5i)12-s − 86.0i·13-s + (−303. − 119. i)14-s + 31.1·15-s + (−48.1 + 83.3i)16-s + (248. − 143. i)17-s + ⋯
L(s)  = 1  + (0.832 − 1.44i)2-s + (0.267 − 0.154i)3-s + (−0.887 − 1.53i)4-s + (0.387 + 0.223i)5-s − 0.515i·6-s + (−0.147 − 0.988i)7-s − 1.29·8-s + (−0.452 + 0.783i)9-s + (0.645 − 0.372i)10-s + (0.834 + 1.44i)11-s + (−0.475 − 0.274i)12-s − 0.509i·13-s + (−1.55 − 0.610i)14-s + 0.138·15-s + (−0.187 + 0.325i)16-s + (0.859 − 0.496i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.481 + 0.876i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.15653 - 1.95373i\)
\(L(\frac12)\) \(\approx\) \(1.15653 - 1.95373i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.68 - 5.59i)T \)
7 \( 1 + (7.25 + 48.4i)T \)
good2 \( 1 + (-3.33 + 5.77i)T + (-8 - 13.8i)T^{2} \)
3 \( 1 + (-2.41 + 1.39i)T + (40.5 - 70.1i)T^{2} \)
11 \( 1 + (-100. - 174. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 86.0iT - 2.85e4T^{2} \)
17 \( 1 + (-248. + 143. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-98.7 - 56.9i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (200. - 347. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 151.T + 7.07e5T^{2} \)
31 \( 1 + (1.28e3 - 740. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (516. - 894. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 940. iT - 2.82e6T^{2} \)
43 \( 1 - 60.1T + 3.41e6T^{2} \)
47 \( 1 + (3.72e3 + 2.14e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-1.71e3 - 2.97e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-4.83e3 + 2.78e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (1.24e3 + 717. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (-1.01e3 - 1.75e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 8.25e3T + 2.54e7T^{2} \)
73 \( 1 + (-7.87e3 + 4.54e3i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (-2.64e3 + 4.58e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 3.23e3iT - 4.74e7T^{2} \)
89 \( 1 + (-6.89e3 - 3.97e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 5.79e3iT - 8.85e7T^{2} \)
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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

−14.65714013703259804218002385162, −13.92917235426441788034547142504, −12.97111094144581961424135641443, −11.80682149695595271801085188887, −10.52251388092492710489470815025, −9.675182041306324421062806586483, −7.36006147706278070054925864564, −5.11062561313551840998124003142, −3.48858708394510733842092799831, −1.72066038085334051954653111446, 3.61165251499963073820432667214, 5.61603430259919156553137010710, 6.38095047251014269062358847155, 8.344881109494949784972002977507, 9.235210870377283735809557040799, 11.70734727895438615604905697734, 12.96796228575782221590839630810, 14.30347772718186736487734126391, 14.74089943973150428631012345946, 16.13670375960793638881746702564

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