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Properties

Label 2-95-19.11-c1-0-7
Degree $2$
Conductor $95$
Sign $-0.0977 + 0.995i$
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  + (1.25 − 2.17i)2-s + (−0.610 + 1.05i)3-s + (−2.14 − 3.71i)4-s + (0.5 − 0.866i)5-s + (1.53 + 2.65i)6-s − 0.221·7-s − 5.72·8-s + (0.753 + 1.30i)9-s + (−1.25 − 2.17i)10-s − 0.778·11-s + 5.23·12-s + (2.5 + 4.33i)13-s + (−0.278 + 0.481i)14-s + (0.610 + 1.05i)15-s + (−2.89 + 5.01i)16-s + (−3.53 + 6.12i)17-s + ⋯
L(s)  = 1  + (0.886 − 1.53i)2-s + (−0.352 + 0.610i)3-s + (−1.07 − 1.85i)4-s + (0.223 − 0.387i)5-s + (0.625 + 1.08i)6-s − 0.0838·7-s − 2.02·8-s + (0.251 + 0.435i)9-s + (−0.396 − 0.686i)10-s − 0.234·11-s + 1.51·12-s + (0.693 + 1.20i)13-s + (−0.0743 + 0.128i)14-s + (0.157 + 0.273i)15-s + (−0.724 + 1.25i)16-s + (−0.858 + 1.48i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.0977 + 0.995i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885059 - 0.976236i\)
\(L(\frac12)\) \(\approx\) \(0.885059 - 0.976236i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-1.33 + 4.15i)T \)
good2 \( 1 + (-1.25 + 2.17i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.610 - 1.05i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 + 0.221T + 7T^{2} \)
11 \( 1 + 0.778T + 11T^{2} \)
13 \( 1 + (-2.5 - 4.33i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.53 - 6.12i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.03 + 6.99i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.110 + 0.192i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.50T + 31T^{2} \)
37 \( 1 + 1.90T + 37T^{2} \)
41 \( 1 + (-3.61 + 6.26i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.64 - 6.32i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.39 - 2.41i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.19 + 3.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.39 + 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.29 - 10.8i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.28 - 9.15i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.92 + 8.52i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.03 + 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.792 - 1.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9.52T + 83T^{2} \)
89 \( 1 + (1.57 + 2.71i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.18 + 5.51i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.39157746608692437214130990871, −12.68779163839996319687031834068, −11.47140667935580545391272989470, −10.75712307708180416296798796212, −9.893160953475731411875340027558, −8.691937427410666075661170750284, −6.22732996632315997922883676864, −4.78569983863991198001990804865, −4.04173055923011594897997732339, −2.04522464518136224795658435750, 3.55518025306837040854492094239, 5.32207301664856979481959953213, 6.22892466159315624691837918229, 7.19607234061139263411013261244, 8.116445647001615071473014538779, 9.725129541917121374380978002045, 11.49504677953085544926938869119, 12.66064059593107750104526973319, 13.45348343955888656095469451880, 14.18863810242564998598743115527

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