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Properties

Label 2-35-7.2-c5-0-0
Degree $2$
Conductor $35$
Sign $-0.765 - 0.642i$
Analytic cond. $5.61343$
Root an. cond. $2.36926$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.56 + 2.70i)2-s + (−13.9 − 24.1i)3-s + (11.1 + 19.2i)4-s + (12.5 − 21.6i)5-s + 86.9·6-s + (−89.4 + 93.8i)7-s − 169.·8-s + (−266. + 462. i)9-s + (39.0 + 67.5i)10-s + (202. + 350. i)11-s + (310. − 537. i)12-s − 931.·13-s + (−113. − 388. i)14-s − 696.·15-s + (−92.0 + 159. i)16-s + (368. + 637. i)17-s + ⋯
L(s)  = 1  + (−0.275 + 0.477i)2-s + (−0.893 − 1.54i)3-s + (0.347 + 0.602i)4-s + (0.223 − 0.387i)5-s + 0.986·6-s + (−0.690 + 0.723i)7-s − 0.935·8-s + (−1.09 + 1.90i)9-s + (0.123 + 0.213i)10-s + (0.504 + 0.874i)11-s + (0.621 − 1.07i)12-s − 1.52·13-s + (−0.155 − 0.529i)14-s − 0.799·15-s + (−0.0898 + 0.155i)16-s + (0.308 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.765 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.765 - 0.642i$
Analytic conductor: \(5.61343\)
Root analytic conductor: \(2.36926\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :5/2),\ -0.765 - 0.642i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.103124 + 0.283260i\)
\(L(\frac12)\) \(\approx\) \(0.103124 + 0.283260i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-12.5 + 21.6i)T \)
7 \( 1 + (89.4 - 93.8i)T \)
good2 \( 1 + (1.56 - 2.70i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (13.9 + 24.1i)T + (-121.5 + 210. i)T^{2} \)
11 \( 1 + (-202. - 350. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 931.T + 3.71e5T^{2} \)
17 \( 1 + (-368. - 637. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (-497. + 861. i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (612. - 1.06e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 5.58e3T + 2.05e7T^{2} \)
31 \( 1 + (1.57e3 + 2.72e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-4.80e3 + 8.32e3i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 1.46e4T + 1.15e8T^{2} \)
43 \( 1 - 3.52e3T + 1.47e8T^{2} \)
47 \( 1 + (9.28e3 - 1.60e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-2.72e3 - 4.72e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-3.88e3 - 6.72e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (9.43e3 - 1.63e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-3.33e4 - 5.77e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.09e4T + 1.80e9T^{2} \)
73 \( 1 + (1.86e4 + 3.23e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-7.83e3 + 1.35e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 7.00e4T + 3.93e9T^{2} \)
89 \( 1 + (-3.95e3 + 6.84e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + 1.49e5T + 8.58e9T^{2} \)
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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

−16.46378514739921392283136912483, −14.93535825914838065101882268043, −13.02640165007210797865854669166, −12.38967020455771743874770049702, −11.67708489326912912281824738880, −9.393566245206076079233397189079, −7.70553342484961873046485554341, −6.83728946008923109419723446980, −5.62937309756088937356110482589, −2.19533805639479081504717967869, 0.20003261744376625484464105443, 3.38897258425981572424914588837, 5.27378956617779844424611851697, 6.58140501690067123646488914582, 9.531909769227743469619424942340, 10.03167784852938086658617628349, 11.01274943092066987044584100544, 11.99243422507947329128842622531, 14.24324036457853699374360891464, 15.18408164181233651052953038772

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