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Properties

Label 2-97-97.96-c5-0-38
Degree $2$
Conductor $97$
Sign $0.735 + 0.677i$
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  + 10.8·2-s + 0.198·3-s + 84.6·4-s − 61.8i·5-s + 2.14·6-s − 122. i·7-s + 569.·8-s − 242.·9-s − 668. i·10-s + 227.·11-s + 16.8·12-s + 528. i·13-s − 1.32e3i·14-s − 12.2i·15-s + 3.43e3·16-s − 1.39e3i·17-s + ⋯
L(s)  = 1  + 1.90·2-s + 0.0127·3-s + 2.64·4-s − 1.10i·5-s + 0.0243·6-s − 0.944i·7-s + 3.14·8-s − 0.999·9-s − 2.11i·10-s + 0.566·11-s + 0.0336·12-s + 0.866i·13-s − 1.80i·14-s − 0.0140i·15-s + 3.35·16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.735 + 0.677i$
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: \((2,\ 97,\ (\ :5/2),\ 0.735 + 0.677i)\)

Particular Values

\(L(3)\) \(\approx\) \(5.32742 - 2.08028i\)
\(L(\frac12)\) \(\approx\) \(5.32742 - 2.08028i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (6.81e4 + 6.27e4i)T \)
good2 \( 1 - 10.8T + 32T^{2} \)
3 \( 1 - 0.198T + 243T^{2} \)
5 \( 1 + 61.8iT - 3.12e3T^{2} \)
7 \( 1 + 122. iT - 1.68e4T^{2} \)
11 \( 1 - 227.T + 1.61e5T^{2} \)
13 \( 1 - 528. iT - 3.71e5T^{2} \)
17 \( 1 + 1.39e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.83e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.31e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.26e3iT - 2.05e7T^{2} \)
31 \( 1 - 7.56e3T + 2.86e7T^{2} \)
37 \( 1 - 3.17e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.17e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.38e4T + 1.47e8T^{2} \)
47 \( 1 + 2.76e4T + 2.29e8T^{2} \)
53 \( 1 + 9.10e3T + 4.18e8T^{2} \)
59 \( 1 + 3.79e4iT - 7.14e8T^{2} \)
61 \( 1 - 5.81e3T + 8.44e8T^{2} \)
67 \( 1 - 2.52e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.09e4iT - 1.80e9T^{2} \)
73 \( 1 - 2.29e4T + 2.07e9T^{2} \)
79 \( 1 + 5.62e4T + 3.07e9T^{2} \)
83 \( 1 + 3.47e4iT - 3.93e9T^{2} \)
89 \( 1 + 5.73e4T + 5.58e9T^{2} \)
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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

−13.08978061278581789297472756137, −11.80971697223610528953684722270, −11.55731324081786788318085840110, −9.821474916677047563780315558073, −8.056100474205281250227713178708, −6.70857853546563732550394049798, −5.50757246640260142420750975770, −4.48738374985960042041416739438, −3.40882721477288278958114792996, −1.48757450570567754830383020683, 2.50720756451278875385227850479, 3.16766548461801408928646330173, 4.84222900464394547481899216604, 6.10037432607414339293882741693, 6.71143199002454706464074501553, 8.419135859081114344824923555838, 10.55273087650971993478520364360, 11.30716070719533850730115678066, 12.20639292265666749915604881914, 13.18109294411899549683627661408

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