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Properties

Label 2-97-97.96-c5-0-31
Degree $2$
Conductor $97$
Sign $0.335 + 0.942i$
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  + 1.99·2-s + 29.1·3-s − 28.0·4-s − 72.5i·5-s + 58.0·6-s − 11.3i·7-s − 119.·8-s + 604.·9-s − 144. i·10-s − 24.9·11-s − 815.·12-s − 833. i·13-s − 22.6i·14-s − 2.11e3i·15-s + 657.·16-s − 1.84e3i·17-s + ⋯
L(s)  = 1  + 0.352·2-s + 1.86·3-s − 0.875·4-s − 1.29i·5-s + 0.658·6-s − 0.0877i·7-s − 0.661·8-s + 2.48·9-s − 0.458i·10-s − 0.0621·11-s − 1.63·12-s − 1.36i·13-s − 0.0309i·14-s − 2.42i·15-s + 0.642·16-s − 1.54i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.335 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.335 + 0.942i$
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: \((2,\ 97,\ (\ :5/2),\ 0.335 + 0.942i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.64126 - 1.86405i\)
\(L(\frac12)\) \(\approx\) \(2.64126 - 1.86405i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (3.10e4 + 8.73e4i)T \)
good2 \( 1 - 1.99T + 32T^{2} \)
3 \( 1 - 29.1T + 243T^{2} \)
5 \( 1 + 72.5iT - 3.12e3T^{2} \)
7 \( 1 + 11.3iT - 1.68e4T^{2} \)
11 \( 1 + 24.9T + 1.61e5T^{2} \)
13 \( 1 + 833. iT - 3.71e5T^{2} \)
17 \( 1 + 1.84e3iT - 1.41e6T^{2} \)
19 \( 1 - 2.65e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.33e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.98e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.54e3T + 2.86e7T^{2} \)
37 \( 1 + 859. iT - 6.93e7T^{2} \)
41 \( 1 - 1.01e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.73e4T + 1.47e8T^{2} \)
47 \( 1 + 4.17e3T + 2.29e8T^{2} \)
53 \( 1 + 3.30e4T + 4.18e8T^{2} \)
59 \( 1 - 9.23e3iT - 7.14e8T^{2} \)
61 \( 1 + 5.34e3T + 8.44e8T^{2} \)
67 \( 1 - 5.56e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.65e4iT - 1.80e9T^{2} \)
73 \( 1 - 8.14e4T + 2.07e9T^{2} \)
79 \( 1 - 8.89e3T + 3.07e9T^{2} \)
83 \( 1 - 1.82e4iT - 3.93e9T^{2} \)
89 \( 1 + 8.26e4T + 5.58e9T^{2} \)
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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

−12.81853131587859747240879186962, −12.56374139121718356119304454083, −10.05375784699613706763423988171, −9.244094273109839593602156240551, −8.434936822191692188737774710288, −7.71138852835235619807432461752, −5.29552344441614478626318505930, −4.16453210832366646482969635608, −2.98118187554157336632843109270, −1.02731345063525155712577419788, 2.17322835975379120552759473206, 3.38461646158873532017970258055, 4.31341375205296379291957439263, 6.56595222502867235590995321805, 7.73997280493020709396158855578, 8.915986723756450073115612307653, 9.568213488203851432289965398094, 10.86306570222454239575297711135, 12.59809260158485005925344581824, 13.70242671767252611329637786676

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