Properties

Label 2-95-95.64-c1-0-6
Degree $2$
Conductor $95$
Sign $0.776 + 0.629i$
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  + (1.73 − i)2-s + (0.999 − 1.73i)4-s + (−1.23 + 1.86i)5-s − 4i·7-s + (−1.5 + 2.59i)9-s + (−0.267 + 4.46i)10-s − 11-s + (−1.73 − i)13-s + (−4 − 6.92i)14-s + (1.99 + 3.46i)16-s + (−1.73 + i)17-s + 6i·18-s + (3.5 − 2.59i)19-s + (2.00 + 3.99i)20-s + (−1.73 + i)22-s + (5.19 + 3i)23-s + ⋯
L(s)  = 1  + (1.22 − 0.707i)2-s + (0.499 − 0.866i)4-s + (−0.550 + 0.834i)5-s − 1.51i·7-s + (−0.5 + 0.866i)9-s + (−0.0847 + 1.41i)10-s − 0.301·11-s + (−0.480 − 0.277i)13-s + (−1.06 − 1.85i)14-s + (0.499 + 0.866i)16-s + (−0.420 + 0.242i)17-s + 1.41i·18-s + (0.802 − 0.596i)19-s + (0.447 + 0.894i)20-s + (−0.369 + 0.213i)22-s + (1.08 + 0.625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.776 + 0.629i$
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: \((2,\ 95,\ (\ :1/2),\ 0.776 + 0.629i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.47247 - 0.522063i\)
\(L(\frac12)\) \(\approx\) \(1.47247 - 0.522063i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.23 - 1.86i)T \)
19 \( 1 + (-3.5 + 2.59i)T \)
good2 \( 1 + (-1.73 + i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (1.5 - 2.59i)T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.19 + 3i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.46 - 2i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.5 - 7.79i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.66 - 5i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + (5.5 - 9.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.19 - 3i)T + (48.5 - 84.0i)T^{2} \)
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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

−13.69537377947042404229149742713, −13.13014767619416361439100889201, −11.61161406012825730199082695579, −11.00314275521396837692234355591, −10.19169022711037439401446761241, −7.997997415445342114601443402482, −6.99686756585096186919380791256, −5.20363376167158838329305437717, −3.99573480617704466686026911245, −2.76886642020995138513698274223, 3.23549390921444222657453093382, 4.88992968633206983173756073625, 5.64913526436897972261373285785, 6.98308444319837001016377361472, 8.537086427225120139224986603788, 9.398209563299963851999341288282, 11.56276094644105126112187956247, 12.35550681774671572464492698213, 12.89047433874651951908672619865, 14.40366878295641971507909120275

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