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Properties

Label 2-97-97.96-c5-0-3
Degree $2$
Conductor $97$
Sign $-0.994 - 0.105i$
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  − 4.70·2-s + 18.5·3-s − 9.83·4-s + 69.5i·5-s − 87.4·6-s + 119. i·7-s + 196.·8-s + 101.·9-s − 327. i·10-s − 565.·11-s − 182.·12-s − 521. i·13-s − 563. i·14-s + 1.29e3i·15-s − 612.·16-s − 375. i·17-s + ⋯
L(s)  = 1  − 0.832·2-s + 1.19·3-s − 0.307·4-s + 1.24i·5-s − 0.991·6-s + 0.922i·7-s + 1.08·8-s + 0.418·9-s − 1.03i·10-s − 1.40·11-s − 0.366·12-s − 0.856i·13-s − 0.768i·14-s + 1.48i·15-s − 0.598·16-s − 0.314i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(97\)
Sign: $-0.994 - 0.105i$
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: \((2,\ 97,\ (\ :5/2),\ -0.994 - 0.105i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0315155 + 0.595960i\)
\(L(\frac12)\) \(\approx\) \(0.0315155 + 0.595960i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad97 \( 1 + (-9.21e4 - 9.77e3i)T \)
good2 \( 1 + 4.70T + 32T^{2} \)
3 \( 1 - 18.5T + 243T^{2} \)
5 \( 1 - 69.5iT - 3.12e3T^{2} \)
7 \( 1 - 119. iT - 1.68e4T^{2} \)
11 \( 1 + 565.T + 1.61e5T^{2} \)
13 \( 1 + 521. iT - 3.71e5T^{2} \)
17 \( 1 + 375. iT - 1.41e6T^{2} \)
19 \( 1 - 732. iT - 2.47e6T^{2} \)
23 \( 1 + 2.83e3iT - 6.43e6T^{2} \)
29 \( 1 - 5.82e3iT - 2.05e7T^{2} \)
31 \( 1 + 9.32e3T + 2.86e7T^{2} \)
37 \( 1 - 9.18e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.24e4iT - 1.15e8T^{2} \)
43 \( 1 + 9.40e3T + 1.47e8T^{2} \)
47 \( 1 + 3.99e3T + 2.29e8T^{2} \)
53 \( 1 + 8.58e3T + 4.18e8T^{2} \)
59 \( 1 - 5.06e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.54e4T + 8.44e8T^{2} \)
67 \( 1 - 3.85e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.64e4iT - 1.80e9T^{2} \)
73 \( 1 + 2.99e3T + 2.07e9T^{2} \)
79 \( 1 - 9.14e4T + 3.07e9T^{2} \)
83 \( 1 - 7.74e4iT - 3.93e9T^{2} \)
89 \( 1 + 7.45e4T + 5.58e9T^{2} \)
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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

−13.66404877744880955538406934897, −12.65129879694033078500658546643, −10.82739512376303326997501282206, −10.16479430571520282087850473996, −8.943045775663037458875160914594, −8.179120502761933654431805890105, −7.26411862481531906974376642465, −5.37439944105885189717478117732, −3.25724797631856903043177783667, −2.30626933928126007867713148022, 0.25970535160416462362791688546, 1.81825460728254989283777153294, 3.87375869810610673430736260296, 5.07918751789410521762860056567, 7.52270266302436107199581214412, 8.152201781559045435079532677525, 9.118460502636746267360902784568, 9.781546652685690229931954095411, 11.10495722470191851190183553314, 13.02588307807949602844182423076

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