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Properties

Label 2-35-35.13-c7-0-1
Degree $2$
Conductor $35$
Sign $0.649 + 0.760i$
Analytic cond. $10.9334$
Root an. cond. $3.30658$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−10.4 + 10.4i)2-s + (−15.5 + 15.5i)3-s − 92.4i·4-s + (−116. + 253. i)5-s − 327. i·6-s + (−900. + 109. i)7-s + (−373. − 373. i)8-s + 1.70e3i·9-s + (−1.43e3 − 3.89e3i)10-s − 2.73e3·11-s + (1.44e3 + 1.44e3i)12-s + (1.44e3 − 1.44e3i)13-s + (8.30e3 − 1.06e4i)14-s + (−2.13e3 − 5.78e3i)15-s + 1.96e4·16-s + (9.76e3 + 9.76e3i)17-s + ⋯
L(s)  = 1  + (−0.927 + 0.927i)2-s + (−0.333 + 0.333i)3-s − 0.722i·4-s + (−0.418 + 0.908i)5-s − 0.618i·6-s + (−0.992 + 0.120i)7-s + (−0.257 − 0.257i)8-s + 0.777i·9-s + (−0.454 − 1.23i)10-s − 0.620·11-s + (0.240 + 0.240i)12-s + (0.182 − 0.182i)13-s + (0.808 − 1.03i)14-s + (−0.163 − 0.442i)15-s + 1.20·16-s + (0.482 + 0.482i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.649 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.649 + 0.760i$
Analytic conductor: \(10.9334\)
Root analytic conductor: \(3.30658\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :7/2),\ 0.649 + 0.760i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0431745 - 0.0199059i\)
\(L(\frac12)\) \(\approx\) \(0.0431745 - 0.0199059i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (116. - 253. i)T \)
7 \( 1 + (900. - 109. i)T \)
good2 \( 1 + (10.4 - 10.4i)T - 128iT^{2} \)
3 \( 1 + (15.5 - 15.5i)T - 2.18e3iT^{2} \)
11 \( 1 + 2.73e3T + 1.94e7T^{2} \)
13 \( 1 + (-1.44e3 + 1.44e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-9.76e3 - 9.76e3i)T + 4.10e8iT^{2} \)
19 \( 1 - 5.45e4T + 8.93e8T^{2} \)
23 \( 1 + (4.51e4 + 4.51e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 1.55e5iT - 1.72e10T^{2} \)
31 \( 1 + 7.87e4iT - 2.75e10T^{2} \)
37 \( 1 + (1.76e5 - 1.76e5i)T - 9.49e10iT^{2} \)
41 \( 1 + 2.13e5iT - 1.94e11T^{2} \)
43 \( 1 + (5.45e5 + 5.45e5i)T + 2.71e11iT^{2} \)
47 \( 1 + (-5.55e5 - 5.55e5i)T + 5.06e11iT^{2} \)
53 \( 1 + (-8.74e4 - 8.74e4i)T + 1.17e12iT^{2} \)
59 \( 1 + 3.12e6T + 2.48e12T^{2} \)
61 \( 1 + 2.97e6iT - 3.14e12T^{2} \)
67 \( 1 + (7.56e4 - 7.56e4i)T - 6.06e12iT^{2} \)
71 \( 1 - 2.57e6T + 9.09e12T^{2} \)
73 \( 1 + (2.68e6 - 2.68e6i)T - 1.10e13iT^{2} \)
79 \( 1 - 2.01e6iT - 1.92e13T^{2} \)
83 \( 1 + (5.18e5 - 5.18e5i)T - 2.71e13iT^{2} \)
89 \( 1 + 3.28e6T + 4.42e13T^{2} \)
97 \( 1 + (7.86e6 + 7.86e6i)T + 8.07e13iT^{2} \)
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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

−15.98440242433648446413458618528, −15.51887596001163947755893251994, −13.95287541022835003786192124536, −12.29114819003493947502599987935, −10.63007640920759682525281203760, −9.777901550812219072650722889121, −8.105490775143324013653056204216, −7.07256439106815700277135331013, −5.72123858797231397190987502945, −3.25205001434808898840224537875, 0.03612673326931727405920756244, 1.12525881022993748799616328466, 3.32357685820582145536090508175, 5.68668988876980172417214256318, 7.64092267304467736361370398007, 9.188671263445484579797078387176, 9.933889297000801339642291558726, 11.65987810460912770203065855612, 12.22493158930365958390404370940, 13.53012430415183907814135127014

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