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Properties

Label 2-95-95.64-c1-0-0
Degree $2$
Conductor $95$
Sign $-0.616 - 0.787i$
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  + (−0.408 + 0.235i)2-s + (−0.900 + 0.520i)3-s + (−0.888 + 1.53i)4-s + (−2.19 + 0.441i)5-s + (0.245 − 0.425i)6-s + 1.17i·7-s − 1.78i·8-s + (−0.958 + 1.66i)9-s + (0.791 − 0.697i)10-s + 0.713·11-s − 1.84i·12-s + (3.55 + 2.05i)13-s + (−0.277 − 0.480i)14-s + (1.74 − 1.53i)15-s + (−1.35 − 2.34i)16-s + (2.21 − 1.27i)17-s + ⋯
L(s)  = 1  + (−0.288 + 0.166i)2-s + (−0.520 + 0.300i)3-s + (−0.444 + 0.769i)4-s + (−0.980 + 0.197i)5-s + (0.100 − 0.173i)6-s + 0.444i·7-s − 0.630i·8-s + (−0.319 + 0.553i)9-s + (0.250 − 0.220i)10-s + 0.215·11-s − 0.533i·12-s + (0.985 + 0.569i)13-s + (−0.0741 − 0.128i)14-s + (0.450 − 0.397i)15-s + (−0.339 − 0.587i)16-s + (0.536 − 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.616 - 0.787i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.616 - 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.220733 + 0.453373i\)
\(L(\frac12)\) \(\approx\) \(0.220733 + 0.453373i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.19 - 0.441i)T \)
19 \( 1 + (1.57 - 4.06i)T \)
good2 \( 1 + (0.408 - 0.235i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.900 - 0.520i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.17iT - 7T^{2} \)
11 \( 1 - 0.713T + 11T^{2} \)
13 \( 1 + (-3.55 - 2.05i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.21 + 1.27i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.525 + 0.303i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.429 - 0.744i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 - 9.38iT - 37T^{2} \)
41 \( 1 + (-2.06 - 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.76 - 5.06i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.10 + 5.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.31 - 2.49i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.12 + 5.41i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.27 + 3.94i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.48 - 2.59i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.58 - 11.4i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-10.7 + 6.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.98 + 5.16i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (-7.98 + 13.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-14.4 + 8.35i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.41890961531860823096327044882, −13.22790560714308596303141267766, −11.96967263007678085465186832488, −11.43612491431986885471688410580, −10.06325299596310332501456090123, −8.590875816378528460836492695291, −7.947091806179220000452029052619, −6.45551082107079726207216603248, −4.74260394039830275953917217723, −3.45572485253898514512929527837, 0.75059016209193079587558431047, 3.84156968303200183015848936717, 5.39601587665444070517928568420, 6.66601061277074235464107247511, 8.182389968372381059581794355231, 9.183386919018027319989275247412, 10.63621438976073441348248685634, 11.32127856973195251769658362895, 12.43775501506260292464126841803, 13.56281536764645965801502103545

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