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Properties

Label 2-35-35.32-c2-0-0
Degree $2$
Conductor $35$
Sign $-0.586 - 0.810i$
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  + (−0.792 + 2.95i)2-s + (0.633 + 2.36i)3-s + (−4.66 − 2.69i)4-s + (−0.307 − 4.99i)5-s − 7.49·6-s + (3.94 + 5.78i)7-s + (2.99 − 2.99i)8-s + (2.60 − 1.50i)9-s + (15.0 + 3.04i)10-s + (−3.12 + 5.40i)11-s + (3.40 − 12.7i)12-s + (11.7 − 11.7i)13-s + (−20.2 + 7.07i)14-s + (11.6 − 3.88i)15-s + (−4.28 − 7.41i)16-s + (−4.56 + 1.22i)17-s + ⋯
L(s)  = 1  + (−0.396 + 1.47i)2-s + (0.211 + 0.788i)3-s + (−1.16 − 0.672i)4-s + (−0.0614 − 0.998i)5-s − 1.24·6-s + (0.563 + 0.826i)7-s + (0.373 − 0.373i)8-s + (0.289 − 0.167i)9-s + (1.50 + 0.304i)10-s + (−0.283 + 0.491i)11-s + (0.284 − 1.06i)12-s + (0.902 − 0.902i)13-s + (−1.44 + 0.505i)14-s + (0.773 − 0.259i)15-s + (−0.267 − 0.463i)16-s + (−0.268 + 0.0720i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.586 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.586 - 0.810i$
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: \((2,\ 35,\ (\ :1),\ -0.586 - 0.810i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.414079 + 0.810734i\)
\(L(\frac12)\) \(\approx\) \(0.414079 + 0.810734i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.307 + 4.99i)T \)
7 \( 1 + (-3.94 - 5.78i)T \)
good2 \( 1 + (0.792 - 2.95i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-0.633 - 2.36i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (3.12 - 5.40i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.7 + 11.7i)T - 169iT^{2} \)
17 \( 1 + (4.56 - 1.22i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-11.4 + 6.58i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (34.8 + 9.33i)T + (458. + 264.5i)T^{2} \)
29 \( 1 + 28.5iT - 841T^{2} \)
31 \( 1 + (-10.1 + 17.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (6.06 - 22.6i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 45.1T + 1.68e3T^{2} \)
43 \( 1 + (21.9 - 21.9i)T - 1.84e3iT^{2} \)
47 \( 1 + (-2.79 + 10.4i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-19.2 - 71.6i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (-14.6 - 8.43i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.7 - 29.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (95.8 - 25.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (26.7 + 99.8i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-36.1 + 20.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-7.39 + 7.39i)T - 6.88e3iT^{2} \)
89 \( 1 + (-19.2 + 11.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (73.6 + 73.6i)T + 9.40e3iT^{2} \)
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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

−16.39585697104912727608699430066, −15.52945115591162364433035034062, −15.15874497089442275969243066956, −13.53816505073181165987344788008, −11.97372683309546009390019614764, −9.914651719244861029655087994854, −8.777663940059113750277064309863, −7.929853799719692761268827910259, −5.88846328776114819756029134061, −4.64814450552333464534840111571, 1.75863358110988447320075688335, 3.69951014259765329092843410493, 6.82952076287258781032707666198, 8.253356787555659683390605029280, 10.07930450017505523462547652075, 10.98872638697917412131188201117, 11.93256202124184509038358033951, 13.47317632094574600288054564448, 14.07222738490866780888809133561, 16.03393619638249560020914903835

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