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Properties

Label 2-95-19.18-c10-0-0
Degree $2$
Conductor $95$
Sign $0.530 - 0.847i$
Analytic cond. $60.3589$
Root an. cond. $7.76910$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 49.4i·2-s + 10.8i·3-s − 1.42e3·4-s − 1.39e3·5-s + 536.·6-s + 1.32e4·7-s + 1.97e4i·8-s + 5.89e4·9-s + 6.91e4i·10-s + 1.34e5·11-s − 1.54e4i·12-s + 7.40e4i·13-s − 6.57e5i·14-s − 1.51e4i·15-s − 4.79e5·16-s − 2.59e6·17-s + ⋯
L(s)  = 1  − 1.54i·2-s + 0.0446i·3-s − 1.39·4-s − 0.447·5-s + 0.0689·6-s + 0.790·7-s + 0.603i·8-s + 0.998·9-s + 0.691i·10-s + 0.833·11-s − 0.0620i·12-s + 0.199i·13-s − 1.22i·14-s − 0.0199i·15-s − 0.457·16-s − 1.82·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(60.3589\)
Root analytic conductor: \(7.76910\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (56, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :5),\ 0.530 - 0.847i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.0885733 + 0.0490639i\)
\(L(\frac12)\) \(\approx\) \(0.0885733 + 0.0490639i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 1.39e3T \)
19 \( 1 + (1.31e6 - 2.09e6i)T \)
good2 \( 1 + 49.4iT - 1.02e3T^{2} \)
3 \( 1 - 10.8iT - 5.90e4T^{2} \)
7 \( 1 - 1.32e4T + 2.82e8T^{2} \)
11 \( 1 - 1.34e5T + 2.59e10T^{2} \)
13 \( 1 - 7.40e4iT - 1.37e11T^{2} \)
17 \( 1 + 2.59e6T + 2.01e12T^{2} \)
23 \( 1 + 9.76e6T + 4.14e13T^{2} \)
29 \( 1 + 2.07e7iT - 4.20e14T^{2} \)
31 \( 1 + 2.24e7iT - 8.19e14T^{2} \)
37 \( 1 + 4.45e7iT - 4.80e15T^{2} \)
41 \( 1 - 1.42e8iT - 1.34e16T^{2} \)
43 \( 1 + 1.33e8T + 2.16e16T^{2} \)
47 \( 1 - 1.20e8T + 5.25e16T^{2} \)
53 \( 1 - 6.01e8iT - 1.74e17T^{2} \)
59 \( 1 + 1.00e9iT - 5.11e17T^{2} \)
61 \( 1 + 4.94e8T + 7.13e17T^{2} \)
67 \( 1 - 2.24e9iT - 1.82e18T^{2} \)
71 \( 1 + 3.66e8iT - 3.25e18T^{2} \)
73 \( 1 + 2.25e9T + 4.29e18T^{2} \)
79 \( 1 - 2.70e9iT - 9.46e18T^{2} \)
83 \( 1 - 2.01e9T + 1.55e19T^{2} \)
89 \( 1 + 4.24e9iT - 3.11e19T^{2} \)
97 \( 1 - 1.23e10iT - 7.37e19T^{2} \)
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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

−11.87654284325741499130865740590, −11.25154214303711440723423515210, −10.23397676463137863244284233073, −9.238875920768020860817387680249, −8.018368994225264421486544373910, −6.51157529561305381111580996025, −4.33120971866867320430248020151, −4.06982490575212111047286694964, −2.20920098558915573450666731356, −1.41695394645581992328026717274, 0.02478096804278616224623870375, 1.81477466340579667565909941394, 4.14914871788045170866066315443, 4.90001623243981997103633471963, 6.48430940414468695672322783773, 7.13352779091634455906061397938, 8.286204941303003798546889311534, 9.069163705466308847404823883179, 10.69947419629448519070276461042, 11.86364994070234283447106434988

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