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Properties

Label 2-95-95.49-c1-0-0
Degree $2$
Conductor $95$
Sign $0.856 - 0.516i$
Analytic cond. $0.758578$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.12 − 1.22i)2-s + (1.35 + 0.780i)3-s + (2.01 + 3.49i)4-s + (−1.45 + 1.70i)5-s + (−1.91 − 3.32i)6-s + 4.50i·7-s − 4.99i·8-s + (−0.281 − 0.487i)9-s + (5.17 − 1.83i)10-s + 2.19·11-s + 6.29i·12-s + (3.25 − 1.87i)13-s + (5.53 − 9.58i)14-s + (−3.29 + 1.16i)15-s + (−2.09 + 3.63i)16-s + (−0.576 − 0.332i)17-s + ⋯
L(s)  = 1  + (−1.50 − 0.868i)2-s + (0.780 + 0.450i)3-s + (1.00 + 1.74i)4-s + (−0.649 + 0.760i)5-s + (−0.782 − 1.35i)6-s + 1.70i·7-s − 1.76i·8-s + (−0.0938 − 0.162i)9-s + (1.63 − 0.580i)10-s + 0.662·11-s + 1.81i·12-s + (0.902 − 0.521i)13-s + (1.47 − 2.56i)14-s + (−0.849 + 0.300i)15-s + (−0.524 + 0.909i)16-s + (−0.139 − 0.0807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ 0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.559269 + 0.155579i\)
\(L(\frac12)\) \(\approx\) \(0.559269 + 0.155579i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.45 - 1.70i)T \)
19 \( 1 + (3.79 - 2.13i)T \)
good2 \( 1 + (2.12 + 1.22i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.35 - 0.780i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + (-3.25 + 1.87i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.576 + 0.332i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.422 + 0.244i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.79 - 3.11i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.83T + 31T^{2} \)
37 \( 1 + 3.01iT - 37T^{2} \)
41 \( 1 + (0.0362 - 0.0627i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.364 - 0.210i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.34 + 2.51i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.26 - 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.26 + 10.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.53 + 6.11i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.95 + 2.86i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.48 - 6.03i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.56 + 1.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.66 + 9.81i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 15.6iT - 83T^{2} \)
89 \( 1 + (0.668 + 1.15i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.79 + 2.19i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.39920537041960389356487361450, −12.42358338087401937754641670510, −11.69945778389984164989817206861, −10.73356541426212926751357042533, −9.578462255975442074894943705506, −8.654769067658833810330665584810, −8.198109972309405239677349916334, −6.37958602823041773154440623440, −3.53558100298402372647174399946, −2.48362557078448155457709764839, 1.16896047195888895046040648865, 4.19964016437422200919886915340, 6.55153383213725490929494879866, 7.48052057758237673622973848498, 8.321715592165311815444631479575, 9.025195137228160796138962251129, 10.36555354937420577152610354418, 11.38347362485622197691465943731, 13.22273748165101071711365987757, 14.00982225422252755646900137426

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