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Properties

Label 2-95-19.11-c1-0-3
Degree $2$
Conductor $95$
Sign $0.287 - 0.957i$
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  + (−0.548 + 0.950i)2-s + (0.189 − 0.328i)3-s + (0.397 + 0.689i)4-s + (−0.5 + 0.866i)5-s + (0.208 + 0.360i)6-s + 1.89·7-s − 3.06·8-s + (1.42 + 2.47i)9-s + (−0.548 − 0.950i)10-s + 0.134·11-s + 0.301·12-s + (−1.75 − 3.04i)13-s + (−1.03 + 1.79i)14-s + (0.189 + 0.328i)15-s + (0.887 − 1.53i)16-s + (0.830 − 1.43i)17-s + ⋯
L(s)  = 1  + (−0.388 + 0.672i)2-s + (0.109 − 0.189i)3-s + (0.198 + 0.344i)4-s + (−0.223 + 0.387i)5-s + (0.0849 + 0.147i)6-s + 0.715·7-s − 1.08·8-s + (0.476 + 0.824i)9-s + (−0.173 − 0.300i)10-s + 0.0405·11-s + 0.0871·12-s + (−0.487 − 0.843i)13-s + (−0.277 + 0.480i)14-s + (0.0489 + 0.0848i)15-s + (0.221 − 0.384i)16-s + (0.201 − 0.348i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.287 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.287 - 0.957i$
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: \((2,\ 95,\ (\ :1/2),\ 0.287 - 0.957i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.728251 + 0.541882i\)
\(L(\frac12)\) \(\approx\) \(0.728251 + 0.541882i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.10 + 3.81i)T \)
good2 \( 1 + (0.548 - 0.950i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.189 + 0.328i)T + (-1.5 - 2.59i)T^{2} \)
7 \( 1 - 1.89T + 7T^{2} \)
11 \( 1 - 0.134T + 11T^{2} \)
13 \( 1 + (1.75 + 3.04i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.830 + 1.43i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.68 + 4.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.48 + 4.30i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.56T + 31T^{2} \)
37 \( 1 + 1.69T + 37T^{2} \)
41 \( 1 + (5.31 - 9.20i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.25 - 7.36i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.55 - 9.62i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.132 - 0.229i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.44 + 5.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.58 + 7.94i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.47 - 2.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (0.664 - 1.15i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.17 + 5.49i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.733 + 1.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.44T + 83T^{2} \)
89 \( 1 + (4.86 + 8.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.73 - 15.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.42752526400959679074800231018, −13.21876737934056632358172515049, −12.05151892115225309869408625381, −11.06007793221899207841359893602, −9.765192668906679572791900823604, −8.145776399454208932025343072373, −7.71926524087744570592638873788, −6.51440827645734283388967561323, −4.83039531908347741928552941327, −2.74659418026308953269554871505, 1.63166237457043358887318819985, 3.78827791908113804786578006591, 5.45370201816457705828551160740, 7.01295277276713635816245937289, 8.564824933073928753998208465424, 9.585271670426735615358706916564, 10.44773697283084872156620532584, 11.85728655962209504217303357795, 12.12111327957290101059866866900, 13.88054285000329392931953723554

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