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Properties

Label 2-35-35.32-c2-0-4
Degree $2$
Conductor $35$
Sign $0.0897 + 0.995i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.506 − 1.89i)2-s + (−0.703 − 2.62i)3-s + (0.141 + 0.0815i)4-s + (−4.52 + 2.11i)5-s − 5.32·6-s + (6.65 + 2.16i)7-s + (5.76 − 5.76i)8-s + (1.39 − 0.808i)9-s + (1.70 + 9.64i)10-s + (−9.62 + 16.6i)11-s + (0.114 − 0.428i)12-s + (0.957 − 0.957i)13-s + (7.46 − 11.5i)14-s + (8.74 + 10.4i)15-s + (−7.66 − 13.2i)16-s + (−12.3 + 3.31i)17-s + ⋯
L(s)  = 1  + (0.253 − 0.946i)2-s + (−0.234 − 0.874i)3-s + (0.0353 + 0.0203i)4-s + (−0.905 + 0.423i)5-s − 0.887·6-s + (0.951 + 0.308i)7-s + (0.720 − 0.720i)8-s + (0.155 − 0.0897i)9-s + (0.170 + 0.964i)10-s + (−0.874 + 1.51i)11-s + (0.00955 − 0.0356i)12-s + (0.0736 − 0.0736i)13-s + (0.533 − 0.821i)14-s + (0.582 + 0.693i)15-s + (−0.478 − 0.829i)16-s + (−0.728 + 0.195i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0897 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.0897 + 0.995i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.0897 + 0.995i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.823459 - 0.752590i\)
\(L(\frac12)\) \(\approx\) \(0.823459 - 0.752590i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.52 - 2.11i)T \)
7 \( 1 + (-6.65 - 2.16i)T \)
good2 \( 1 + (-0.506 + 1.89i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (0.703 + 2.62i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (9.62 - 16.6i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-0.957 + 0.957i)T - 169iT^{2} \)
17 \( 1 + (12.3 - 3.31i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (4.15 - 2.40i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (10.2 + 2.74i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 13.7iT - 841T^{2} \)
31 \( 1 + (9.56 - 16.5i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (1.12 - 4.21i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 50.7T + 1.68e3T^{2} \)
43 \( 1 + (31.8 - 31.8i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.6 + 43.3i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (15.8 + 59.2i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-73.0 - 42.1i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.64 + 2.85i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-29.4 + 7.90i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 74.6T + 5.04e3T^{2} \)
73 \( 1 + (22.0 + 82.2i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (43.4 - 25.0i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (58.5 - 58.5i)T - 6.88e3iT^{2} \)
89 \( 1 + (-74.6 + 43.1i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-16.3 - 16.3i)T + 9.40e3iT^{2} \)
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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

−15.83918281905181351667659645908, −14.86929463548741456163543040635, −13.09511022119155944383735589266, −12.26869586357875176499542261543, −11.45966834456244347106386290362, −10.29322045639237363828037236242, −7.909251716249540044944206392628, −6.98735808814060929248749853473, −4.38919302259427573298147554432, −2.11956314562525767340582151602, 4.35065358359383984556384506124, 5.49481038229736814763488462371, 7.49880571226567332998181507211, 8.544098355831010485359043636219, 10.76114800067751960483708561528, 11.30469431389186310311414749678, 13.37335717317924729866985620405, 14.63251254235797845762312419487, 15.79266498811675421601271131087, 16.11214883734818787794032350497

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