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Properties

Label 2-97-97.96-c5-0-33
Degree $2$
Conductor $97$
Sign $0.995 + 0.0922i$
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  + 9.70·2-s + 23.5·3-s + 62.0·4-s − 57.7i·5-s + 228.·6-s + 191. i·7-s + 291.·8-s + 312.·9-s − 560. i·10-s − 661.·11-s + 1.46e3·12-s + 473. i·13-s + 1.85e3i·14-s − 1.36e3i·15-s + 845.·16-s − 1.30e3i·17-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.51·3-s + 1.94·4-s − 1.03i·5-s + 2.59·6-s + 1.47i·7-s + 1.61·8-s + 1.28·9-s − 1.77i·10-s − 1.64·11-s + 2.93·12-s + 0.777i·13-s + 2.53i·14-s − 1.56i·15-s + 0.825·16-s − 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.995 + 0.0922i$
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: \((2,\ 97,\ (\ :5/2),\ 0.995 + 0.0922i)\)

Particular Values

\(L(3)\) \(\approx\) \(7.04255 - 0.325464i\)
\(L(\frac12)\) \(\approx\) \(7.04255 - 0.325464i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (9.22e4 + 8.54e3i)T \)
good2 \( 1 - 9.70T + 32T^{2} \)
3 \( 1 - 23.5T + 243T^{2} \)
5 \( 1 + 57.7iT - 3.12e3T^{2} \)
7 \( 1 - 191. iT - 1.68e4T^{2} \)
11 \( 1 + 661.T + 1.61e5T^{2} \)
13 \( 1 - 473. iT - 3.71e5T^{2} \)
17 \( 1 + 1.30e3iT - 1.41e6T^{2} \)
19 \( 1 + 3.09e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.65e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.92e3iT - 2.05e7T^{2} \)
31 \( 1 + 460.T + 2.86e7T^{2} \)
37 \( 1 - 1.03e4iT - 6.93e7T^{2} \)
41 \( 1 + 3.95e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.01e4T + 1.47e8T^{2} \)
47 \( 1 + 9.22e3T + 2.29e8T^{2} \)
53 \( 1 - 3.59e4T + 4.18e8T^{2} \)
59 \( 1 + 2.35e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.68e4T + 8.44e8T^{2} \)
67 \( 1 - 4.60e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.43e3iT - 1.80e9T^{2} \)
73 \( 1 + 2.72e4T + 2.07e9T^{2} \)
79 \( 1 - 3.81e4T + 3.07e9T^{2} \)
83 \( 1 - 3.28e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.25e4T + 5.58e9T^{2} \)
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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

−13.24469586087526318826786828143, −12.43737213646032405620134218612, −11.38789372759942513406102416872, −9.313833029148538880947880868882, −8.625483447235735077320027968376, −7.21947904828776580563947879530, −5.39251995522460459553285334740, −4.70095905750257389615164737940, −2.92377568242152351434070720899, −2.33069221294537702522162187769, 2.31582490494896739095402263340, 3.33758862681327932956415755901, 4.10058383259350950894094831624, 5.90960115972444680970682608933, 7.39337805388158522852188289069, 7.966366427589967351612254399627, 10.38236936672976023240018640377, 10.61491970715538270990074783057, 12.62470774370920618767034826765, 13.29264036083048161876306339842

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