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Properties

Label 2-97-97.96-c5-0-17
Degree $2$
Conductor $97$
Sign $0.117 + 0.993i$
Analytic cond. $15.5572$
Root an. cond. $3.94426$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.13·2-s − 26.9·3-s − 22.1·4-s + 95.4i·5-s − 84.3·6-s + 82.4i·7-s − 169.·8-s + 481.·9-s + 299. i·10-s − 573.·11-s + 596.·12-s − 288. i·13-s + 258. i·14-s − 2.56e3i·15-s + 176.·16-s + 997. i·17-s + ⋯
L(s)  = 1  + 0.554·2-s − 1.72·3-s − 0.692·4-s + 1.70i·5-s − 0.957·6-s + 0.636i·7-s − 0.938·8-s + 1.97·9-s + 0.946i·10-s − 1.42·11-s + 1.19·12-s − 0.473i·13-s + 0.352i·14-s − 2.94i·15-s + 0.172·16-s + 0.837i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(15.5572\)
Root analytic conductor: \(3.94426\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{97} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 97,\ (\ :5/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0274628 - 0.0244120i\)
\(L(\frac12)\) \(\approx\) \(0.0274628 - 0.0244120i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (1.08e4 + 9.20e4i)T \)
good2 \( 1 - 3.13T + 32T^{2} \)
3 \( 1 + 26.9T + 243T^{2} \)
5 \( 1 - 95.4iT - 3.12e3T^{2} \)
7 \( 1 - 82.4iT - 1.68e4T^{2} \)
11 \( 1 + 573.T + 1.61e5T^{2} \)
13 \( 1 + 288. iT - 3.71e5T^{2} \)
17 \( 1 - 997. iT - 1.41e6T^{2} \)
19 \( 1 - 828. iT - 2.47e6T^{2} \)
23 \( 1 + 3.90e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.94e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.31e3T + 2.86e7T^{2} \)
37 \( 1 - 1.53e4iT - 6.93e7T^{2} \)
41 \( 1 - 5.08e3iT - 1.15e8T^{2} \)
43 \( 1 + 4.04e3T + 1.47e8T^{2} \)
47 \( 1 - 6.27e3T + 2.29e8T^{2} \)
53 \( 1 - 63.7T + 4.18e8T^{2} \)
59 \( 1 - 2.40e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.90e4T + 8.44e8T^{2} \)
67 \( 1 + 5.74e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.80e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.69e4T + 2.07e9T^{2} \)
79 \( 1 + 8.65e4T + 3.07e9T^{2} \)
83 \( 1 - 4.54e4iT - 3.93e9T^{2} \)
89 \( 1 - 9.78e4T + 5.58e9T^{2} \)
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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

−12.58673965053006987474225424508, −11.76801970523630801913380033013, −10.50823095449712286397098453804, −10.22571598342581672498420596943, −8.014978797035062746501272175799, −6.40879353099606249986426995773, −5.83052258538587576788145806464, −4.62749911674713214153846209869, −2.87639048606378126262420361604, −0.02128274593900758146575608720, 0.897367267266545594596398586980, 4.27393481743081608919411426768, 5.10237640783234959715966592760, 5.62441049880929851716242968074, 7.43618494236716436221700440766, 9.004603642016281325977991668822, 10.09776400754052691179133485541, 11.42250968402880776298161066219, 12.33146673842283033411504773597, 13.06306677890169838587737029101

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