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Properties

Label 2-35-35.2-c2-0-0
Degree $2$
Conductor $35$
Sign $-0.981 - 0.191i$
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  + (−1.23 + 0.330i)2-s + (−4.01 − 1.07i)3-s + (−2.05 + 1.18i)4-s + (−1.75 + 4.68i)5-s + 5.30·6-s + (−3.92 − 5.79i)7-s + (5.74 − 5.74i)8-s + (7.17 + 4.14i)9-s + (0.613 − 6.34i)10-s + (2.98 + 5.16i)11-s + (9.53 − 2.55i)12-s + (−15.4 + 15.4i)13-s + (6.74 + 5.84i)14-s + (12.0 − 16.9i)15-s + (−0.434 + 0.752i)16-s + (−0.226 + 0.846i)17-s + ⋯
L(s)  = 1  + (−0.615 + 0.165i)2-s + (−1.33 − 0.358i)3-s + (−0.513 + 0.296i)4-s + (−0.350 + 0.936i)5-s + 0.883·6-s + (−0.561 − 0.827i)7-s + (0.718 − 0.718i)8-s + (0.797 + 0.460i)9-s + (0.0613 − 0.634i)10-s + (0.271 + 0.469i)11-s + (0.794 − 0.212i)12-s + (−1.18 + 1.18i)13-s + (0.482 + 0.417i)14-s + (0.805 − 1.12i)15-s + (−0.0271 + 0.0470i)16-s + (−0.0133 + 0.0498i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.981 - 0.191i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.981 - 0.191i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ -0.981 - 0.191i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0102132 + 0.105680i\)
\(L(\frac12)\) \(\approx\) \(0.0102132 + 0.105680i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.75 - 4.68i)T \)
7 \( 1 + (3.92 + 5.79i)T \)
good2 \( 1 + (1.23 - 0.330i)T + (3.46 - 2i)T^{2} \)
3 \( 1 + (4.01 + 1.07i)T + (7.79 + 4.5i)T^{2} \)
11 \( 1 + (-2.98 - 5.16i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (15.4 - 15.4i)T - 169iT^{2} \)
17 \( 1 + (0.226 - 0.846i)T + (-250. - 144.5i)T^{2} \)
19 \( 1 + (18.0 + 10.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-3.12 - 11.6i)T + (-458. + 264.5i)T^{2} \)
29 \( 1 - 7.52iT - 841T^{2} \)
31 \( 1 + (9.90 + 17.1i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (-13.4 + 3.59i)T + (1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 12.0T + 1.68e3T^{2} \)
43 \( 1 + (-12.3 + 12.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (43.9 - 11.7i)T + (1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (9.78 + 2.62i)T + (2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-18.2 + 10.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (15.1 - 26.1i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (18.6 - 69.6i)T + (-3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 128.T + 5.04e3T^{2} \)
73 \( 1 + (-72.5 - 19.4i)T + (4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (14.8 + 8.57i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-4.42 + 4.42i)T - 6.88e3iT^{2} \)
89 \( 1 + (-94.1 - 54.3i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (55.4 + 55.4i)T + 9.40e3iT^{2} \)
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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

−17.12708159290722689738003998822, −16.28024033390778013268412248266, −14.54216657838046174423987678186, −13.08873159997547254567498435681, −11.88892259472447085151961011241, −10.69107260748351409165607119955, −9.540504540672038176802095039831, −7.35939761783109188811526290176, −6.67231583702701603341649651917, −4.34236266481791297715036928340, 0.17737315119154646895456127984, 4.81470122483279353355302024820, 5.83805323029548511702425281962, 8.286247364078865951243285371767, 9.555607926603585463161875545803, 10.66448862078788663352445629677, 12.03309536819280401630489195107, 12.90197024656079060202333551655, 14.90049817072380571401901809086, 16.23511931031580472917869172437

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