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Properties

Label 2-95-19.11-c3-0-12
Degree $2$
Conductor $95$
Sign $0.910 + 0.412i$
Analytic cond. $5.60518$
Root an. cond. $2.36752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (2.5 − 4.33i)3-s + (3.5 + 6.06i)4-s + (−2.5 + 4.33i)5-s + (−2.5 − 4.33i)6-s + 22·7-s + 15·8-s + (0.999 + 1.73i)9-s + (2.5 + 4.33i)10-s + 9·11-s + 35·12-s + (−27 − 46.7i)13-s + (11 − 19.0i)14-s + (12.5 + 21.6i)15-s + (−20.5 + 35.5i)16-s + (27 − 46.7i)17-s + ⋯
L(s)  = 1  + (0.176 − 0.306i)2-s + (0.481 − 0.833i)3-s + (0.437 + 0.757i)4-s + (−0.223 + 0.387i)5-s + (−0.170 − 0.294i)6-s + 1.18·7-s + 0.662·8-s + (0.0370 + 0.0641i)9-s + (0.0790 + 0.136i)10-s + 0.246·11-s + 0.841·12-s + (−0.576 − 0.997i)13-s + (0.209 − 0.363i)14-s + (0.215 + 0.372i)15-s + (−0.320 + 0.554i)16-s + (0.385 − 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.910 + 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.910 + 0.412i$
Analytic conductor: \(5.60518\)
Root analytic conductor: \(2.36752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :3/2),\ 0.910 + 0.412i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.25507 - 0.487383i\)
\(L(\frac12)\) \(\approx\) \(2.25507 - 0.487383i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.5 - 4.33i)T \)
19 \( 1 + (66.5 + 49.3i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-2.5 + 4.33i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 - 22T + 343T^{2} \)
11 \( 1 - 9T + 1.33e3T^{2} \)
13 \( 1 + (27 + 46.7i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + (-27 + 46.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
23 \( 1 + (-46 - 79.6i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-67 - 116. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 252T + 2.97e4T^{2} \)
37 \( 1 + 236T + 5.06e4T^{2} \)
41 \( 1 + (-121.5 + 210. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (248 - 429. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (251 + 434. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (31 + 53.6i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (340.5 - 589. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-71 - 122. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (27.5 + 47.6i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-487 + 843. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + (347.5 - 601. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-368 + 637. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 63T + 5.71e5T^{2} \)
89 \( 1 + (363 + 628. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-583.5 + 1.01e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.27845319861924151200689279822, −12.42966042577659884862406238900, −11.44486107416496694412240750897, −10.56107768258407645578290151821, −8.604153506381280619778995602243, −7.65583369751070102026145787483, −7.06685413320404984814767140528, −4.92045739555757377399174611867, −3.13095906600118597961816098731, −1.81864937723505996313741253251, 1.74699357876445148617334030237, 4.13421292185354867165874738185, 5.05702198807915921117737565279, 6.62523077144777768688396095426, 8.079984804409519744365686719613, 9.218415360922711377934697209765, 10.31272078955377143794714183235, 11.27649951212410368447791092906, 12.44058131781760656095872224115, 14.21712530343059068595670031907

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