Properties

Label 2-95-95.94-c4-0-13
Degree $2$
Conductor $95$
Sign $0.619 - 0.784i$
Analytic cond. $9.82014$
Root an. cond. $3.13371$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·4-s + (15.5 − 19.6i)5-s + 65.3i·7-s − 81·9-s + 233·11-s + 256·16-s + 457. i·17-s + 361·19-s + (−248 + 313. i)20-s + 1.04e3i·23-s + (−144. − 608. i)25-s − 1.04e3i·28-s + (1.28e3 + 1.01e3i)35-s + 1.29e3·36-s + 1.11e3i·43-s − 3.72e3·44-s + ⋯
L(s)  = 1  − 4-s + (0.619 − 0.784i)5-s + 1.33i·7-s − 9-s + 1.92·11-s + 16-s + 1.58i·17-s + 19-s + (−0.619 + 0.784i)20-s + 1.97i·23-s + (−0.231 − 0.972i)25-s − 1.33i·28-s + (1.04 + 0.827i)35-s + 36-s + 0.601i·43-s − 1.92·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(95\)    =    \(5 \cdot 19\)
Sign: $0.619 - 0.784i$
Analytic conductor: \(9.82014\)
Root analytic conductor: \(3.13371\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{95} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 95,\ (\ :2),\ 0.619 - 0.784i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.27595 + 0.617975i\)
\(L(\frac12)\) \(\approx\) \(1.27595 + 0.617975i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-15.5 + 19.6i)T \)
19 \( 1 - 361T \)
good2 \( 1 + 16T^{2} \)
3 \( 1 + 81T^{2} \)
7 \( 1 - 65.3iT - 2.40e3T^{2} \)
11 \( 1 - 233T + 1.46e4T^{2} \)
13 \( 1 + 2.85e4T^{2} \)
17 \( 1 - 457. iT - 8.35e4T^{2} \)
23 \( 1 - 1.04e3iT - 2.79e5T^{2} \)
29 \( 1 - 7.07e5T^{2} \)
31 \( 1 - 9.23e5T^{2} \)
37 \( 1 + 1.87e6T^{2} \)
41 \( 1 - 2.82e6T^{2} \)
43 \( 1 - 1.11e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.24e3iT - 4.87e6T^{2} \)
53 \( 1 + 7.89e6T^{2} \)
59 \( 1 - 1.21e7T^{2} \)
61 \( 1 + 3.16e3T + 1.38e7T^{2} \)
67 \( 1 + 2.01e7T^{2} \)
71 \( 1 - 2.54e7T^{2} \)
73 \( 1 - 3.59e3iT - 2.83e7T^{2} \)
79 \( 1 - 3.89e7T^{2} \)
83 \( 1 - 1.25e4iT - 4.74e7T^{2} \)
89 \( 1 - 6.27e7T^{2} \)
97 \( 1 + 8.85e7T^{2} \)
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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

−13.48559050732049559766878704582, −12.31052597209563439822921729227, −11.63790066229888743099447853155, −9.649954713760270084685920186065, −9.047054067518865911720189175949, −8.350922455075624826772674772696, −6.02915077023168424855130583504, −5.34881228310999129018216680068, −3.68945884836137828924786831286, −1.47765122615129073750647223808, 0.78481408612752177089805692941, 3.24553704052601245298815987119, 4.58558262019155727278335145272, 6.21120762413494093309307430008, 7.33206726668424913589399642107, 8.947560471711389729707366350072, 9.724669212263119667036604910974, 10.87880375862105561651700749361, 11.98278851078587289704862415083, 13.56366994200087445734826103036

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