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Properties

Label 2-97-97.96-c5-0-25
Degree $2$
Conductor $97$
Sign $0.951 + 0.308i$
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  − 8.40·2-s + 27.7·3-s + 38.6·4-s + 56.8i·5-s − 233.·6-s − 175. i·7-s − 55.6·8-s + 528.·9-s − 477. i·10-s + 230.·11-s + 1.07e3·12-s − 567. i·13-s + 1.47e3i·14-s + 1.57e3i·15-s − 768.·16-s − 1.40e3i·17-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.78·3-s + 1.20·4-s + 1.01i·5-s − 2.64·6-s − 1.35i·7-s − 0.307·8-s + 2.17·9-s − 1.51i·10-s + 0.573·11-s + 2.15·12-s − 0.932i·13-s + 2.01i·14-s + 1.81i·15-s − 0.750·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.951 + 0.308i$
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: \((2,\ 97,\ (\ :5/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.69363 - 0.267990i\)
\(L(\frac12)\) \(\approx\) \(1.69363 - 0.267990i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (8.81e4 + 2.86e4i)T \)
good2 \( 1 + 8.40T + 32T^{2} \)
3 \( 1 - 27.7T + 243T^{2} \)
5 \( 1 - 56.8iT - 3.12e3T^{2} \)
7 \( 1 + 175. iT - 1.68e4T^{2} \)
11 \( 1 - 230.T + 1.61e5T^{2} \)
13 \( 1 + 567. iT - 3.71e5T^{2} \)
17 \( 1 + 1.40e3iT - 1.41e6T^{2} \)
19 \( 1 - 204. iT - 2.47e6T^{2} \)
23 \( 1 - 5.06e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.84e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.92e3T + 2.86e7T^{2} \)
37 \( 1 + 2.32e3iT - 6.93e7T^{2} \)
41 \( 1 + 2.12e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.12e4T + 1.47e8T^{2} \)
47 \( 1 - 1.92e4T + 2.29e8T^{2} \)
53 \( 1 - 1.99e4T + 4.18e8T^{2} \)
59 \( 1 + 9.52e3iT - 7.14e8T^{2} \)
61 \( 1 - 4.22e4T + 8.44e8T^{2} \)
67 \( 1 - 6.36e4iT - 1.35e9T^{2} \)
71 \( 1 + 2.83e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.88e4T + 2.07e9T^{2} \)
79 \( 1 + 1.65e4T + 3.07e9T^{2} \)
83 \( 1 - 8.71e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
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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

−13.52970041564995745736213002338, −11.34742403699854329707656156663, −10.10798745103898867356083459812, −9.758083237283043936132070854868, −8.496958762093263041297491091662, −7.44619949879926798442299289450, −7.12506596526346357329217749518, −3.86760149229756970482466110515, −2.66655437525657785433944936710, −1.06528567198981444363008458079, 1.41238083592332626370036209254, 2.45049847992895633247657725125, 4.40282451335759987843134824018, 6.72641604838293329296031944510, 8.315473499994994034936407546321, 8.696044230639999024143731875961, 9.139536960379607344794316013057, 10.26121044474749880485289934949, 12.03808055003020115171319777507, 12.98787861678863814434642174944

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