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Properties

Label 2-35-7.2-c5-0-9
Degree $2$
Conductor $35$
Sign $0.662 + 0.748i$
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  + (3.61 − 6.25i)2-s + (7.96 + 13.7i)3-s + (−10.1 − 17.5i)4-s + (12.5 − 21.6i)5-s + 115.·6-s + (102. − 79.5i)7-s + 85.0·8-s + (−5.23 + 9.07i)9-s + (−90.3 − 156. i)10-s + (193. + 335. i)11-s + (161. − 278. i)12-s − 920.·13-s + (−128. − 928. i)14-s + 398.·15-s + (631. − 1.09e3i)16-s + (−339. − 587. i)17-s + ⋯
L(s)  = 1  + (0.638 − 1.10i)2-s + (0.510 + 0.884i)3-s + (−0.316 − 0.547i)4-s + (0.223 − 0.387i)5-s + 1.30·6-s + (0.789 − 0.613i)7-s + 0.469·8-s + (−0.0215 + 0.0373i)9-s + (−0.285 − 0.494i)10-s + (0.482 + 0.835i)11-s + (0.322 − 0.559i)12-s − 1.51·13-s + (−0.174 − 1.26i)14-s + 0.456·15-s + (0.616 − 1.06i)16-s + (−0.284 − 0.493i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.662 + 0.748i$
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: \((2,\ 35,\ (\ :5/2),\ 0.662 + 0.748i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.48729 - 1.12052i\)
\(L(\frac12)\) \(\approx\) \(2.48729 - 1.12052i\)
\(L(\frac{7}{2})\) 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 \( 1 + (-102. + 79.5i)T \)
good2 \( 1 + (-3.61 + 6.25i)T + (-16 - 27.7i)T^{2} \)
3 \( 1 + (-7.96 - 13.7i)T + (-121.5 + 210. i)T^{2} \)
11 \( 1 + (-193. - 335. i)T + (-8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 + 920.T + 3.71e5T^{2} \)
17 \( 1 + (339. + 587. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (1.38e3 - 2.39e3i)T + (-1.23e6 - 2.14e6i)T^{2} \)
23 \( 1 + (1.03e3 - 1.78e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + 2.92e3T + 2.05e7T^{2} \)
31 \( 1 + (2.12e3 + 3.68e3i)T + (-1.43e7 + 2.47e7i)T^{2} \)
37 \( 1 + (-77.3 + 133. i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + 7.60e3T + 1.15e8T^{2} \)
43 \( 1 - 1.35e4T + 1.47e8T^{2} \)
47 \( 1 + (7.50e3 - 1.29e4i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (9.03e3 + 1.56e4i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (3.23e3 + 5.61e3i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.00e3 - 3.46e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.68e4 - 2.91e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + 6.14e4T + 1.80e9T^{2} \)
73 \( 1 + (-2.13e4 - 3.69e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-5.39e4 + 9.34e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 9.90e4T + 3.93e9T^{2} \)
89 \( 1 + (3.83e3 - 6.63e3i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 8.70e4T + 8.58e9T^{2} \)
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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

−14.85690437572879512857928019406, −14.23216403397857928727742887667, −12.75690121272731030791190754815, −11.76952068717286779552268851421, −10.32792495999473868625404138711, −9.532652632339800559721459709162, −7.61687291688214207952484957292, −4.79866634362436162940332736038, −3.91664410615573750149989761012, −1.92801831662003578357190781798, 2.17369095477785778224360094476, 4.87410092672232946021294375378, 6.43091461951914793900649064197, 7.49667485930098099839301245998, 8.691923150661627833162719909703, 10.85844851300425863966013067078, 12.50963392940994446893133683898, 13.67899029484295705128849667098, 14.51438209826169707550244533171, 15.23659870275721747891844354160

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