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Properties

Label 2-95-5.4-c1-0-0
Degree $2$
Conductor $95$
Sign $-0.929 - 0.369i$
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  + 2.41i·2-s + 0.537i·3-s − 3.85·4-s + (−2.07 − 0.826i)5-s − 1.29·6-s + 3.18i·7-s − 4.49i·8-s + 2.71·9-s + (2 − 5.02i)10-s + 4.15·11-s − 2.07i·12-s + 2.07i·13-s − 7.71·14-s + (0.443 − 1.11i)15-s + 3.15·16-s − 5.79i·17-s + ⋯
L(s)  = 1  + 1.71i·2-s + 0.310i·3-s − 1.92·4-s + (−0.929 − 0.369i)5-s − 0.530·6-s + 1.20i·7-s − 1.58i·8-s + 0.903·9-s + (0.632 − 1.58i)10-s + 1.25·11-s − 0.597i·12-s + 0.574i·13-s − 2.06·14-s + (0.114 − 0.288i)15-s + 0.788·16-s − 1.40i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $-0.929 - 0.369i$
Analytic conductor: \(0.758578\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :1/2),\ -0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.161267 + 0.841718i\)
\(L(\frac12)\) \(\approx\) \(0.161267 + 0.841718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.07 + 0.826i)T \)
19 \( 1 - T \)
good2 \( 1 - 2.41iT - 2T^{2} \)
3 \( 1 - 0.537iT - 3T^{2} \)
7 \( 1 - 3.18iT - 7T^{2} \)
11 \( 1 - 4.15T + 11T^{2} \)
13 \( 1 - 2.07iT - 13T^{2} \)
17 \( 1 + 5.79iT - 17T^{2} \)
23 \( 1 + 2.60iT - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 + 4.30iT - 37T^{2} \)
41 \( 1 + 0.599T + 41T^{2} \)
43 \( 1 - 3.18iT - 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 - 11.7iT - 53T^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 + 8.75T + 61T^{2} \)
67 \( 1 + 4.76iT - 67T^{2} \)
71 \( 1 - 13.7T + 71T^{2} \)
73 \( 1 + 2.72iT - 73T^{2} \)
79 \( 1 - 1.40T + 79T^{2} \)
83 \( 1 + 7.07iT - 83T^{2} \)
89 \( 1 + 16.5T + 89T^{2} \)
97 \( 1 + 2.07iT - 97T^{2} \)
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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

−14.91587755828743996140732734371, −13.84451991241822544180007603552, −12.44442891085560564719219286913, −11.53950145719745279541421919243, −9.354185330099607118717815923817, −8.915038587374742941916063910158, −7.56623712770699265231123577404, −6.63054945188045570305069337188, −5.18809842675337315714788227610, −4.13013001068348509602005960455, 1.29396656358061951707671114762, 3.60313810374584796487520343174, 4.22537240593369455802252818368, 6.84997531671277012796673575455, 8.078641367623309618278074020147, 9.653356311466335909467732732958, 10.56916203529522913710147082227, 11.34510190046378379044315038438, 12.36096432927352159860279984005, 13.12467186518502437031035265611

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