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Properties

Label 2-97-97.96-c5-0-15
Degree $2$
Conductor $97$
Sign $0.507 - 0.861i$
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  − 2.57·2-s + 15.1·3-s − 25.3·4-s + 24.3i·5-s − 39.0·6-s − 87.8i·7-s + 147.·8-s − 12.8·9-s − 62.6i·10-s + 375.·11-s − 385.·12-s + 791. i·13-s + 225. i·14-s + 369. i·15-s + 433.·16-s − 346. i·17-s + ⋯
L(s)  = 1  − 0.454·2-s + 0.973·3-s − 0.793·4-s + 0.435i·5-s − 0.442·6-s − 0.677i·7-s + 0.815·8-s − 0.0528·9-s − 0.197i·10-s + 0.936·11-s − 0.772·12-s + 1.29i·13-s + 0.308i·14-s + 0.423i·15-s + 0.422·16-s − 0.290i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.507 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $0.507 - 0.861i$
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: \((2,\ 97,\ (\ :5/2),\ 0.507 - 0.861i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.33437 + 0.762618i\)
\(L(\frac12)\) \(\approx\) \(1.33437 + 0.762618i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (4.70e4 - 7.98e4i)T \)
good2 \( 1 + 2.57T + 32T^{2} \)
3 \( 1 - 15.1T + 243T^{2} \)
5 \( 1 - 24.3iT - 3.12e3T^{2} \)
7 \( 1 + 87.8iT - 1.68e4T^{2} \)
11 \( 1 - 375.T + 1.61e5T^{2} \)
13 \( 1 - 791. iT - 3.71e5T^{2} \)
17 \( 1 + 346. iT - 1.41e6T^{2} \)
19 \( 1 - 2.18e3iT - 2.47e6T^{2} \)
23 \( 1 + 401. iT - 6.43e6T^{2} \)
29 \( 1 - 8.18e3iT - 2.05e7T^{2} \)
31 \( 1 - 8.23e3T + 2.86e7T^{2} \)
37 \( 1 - 4.63e3iT - 6.93e7T^{2} \)
41 \( 1 - 1.54e4iT - 1.15e8T^{2} \)
43 \( 1 - 1.00e4T + 1.47e8T^{2} \)
47 \( 1 + 1.00e3T + 2.29e8T^{2} \)
53 \( 1 - 1.00e4T + 4.18e8T^{2} \)
59 \( 1 + 4.35e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.99e4T + 8.44e8T^{2} \)
67 \( 1 + 4.57e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.00e4iT - 1.80e9T^{2} \)
73 \( 1 + 5.63e4T + 2.07e9T^{2} \)
79 \( 1 + 7.79e3T + 3.07e9T^{2} \)
83 \( 1 - 8.73e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.33e5T + 5.58e9T^{2} \)
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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

−13.59737173044812175011706595436, −12.13423248052373626542839794094, −10.74516453550421086347927622051, −9.607122681778104849671280587692, −8.855752448376269789284041930177, −7.84499557879090373747023835055, −6.59273735348368704003562190927, −4.50146485556698867694965844543, −3.37098724050211422184719082069, −1.40343217868065727357001371667, 0.73203681721398646777752015685, 2.69347615053871269348817149973, 4.25723215038917387752090754867, 5.70935659890865215462906802503, 7.65362423156223472662338898853, 8.754262813103414025255003425693, 9.041733646863258995656360171664, 10.29856296101961397067969993079, 11.85562400282812806513310116204, 13.06206201704111222212206489164

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