Properties

Label 2-91-91.23-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.968 - 0.250i$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.58i·2-s + (−0.259 + 0.449i)3-s − 4.70·4-s + (−1.39 − 0.806i)5-s + (−1.16 − 0.671i)6-s + (1.06 + 2.42i)7-s − 6.99i·8-s + (1.36 + 2.36i)9-s + (2.08 − 3.61i)10-s + (2.34 + 1.35i)11-s + (1.21 − 2.11i)12-s + (2.36 − 2.71i)13-s + (−6.27 + 2.74i)14-s + (0.723 − 0.417i)15-s + 8.69·16-s − 3.12·17-s + ⋯
L(s)  = 1  + 1.83i·2-s + (−0.149 + 0.259i)3-s − 2.35·4-s + (−0.624 − 0.360i)5-s + (−0.474 − 0.273i)6-s + (0.401 + 0.915i)7-s − 2.47i·8-s + (0.455 + 0.788i)9-s + (0.659 − 1.14i)10-s + (0.706 + 0.407i)11-s + (0.351 − 0.609i)12-s + (0.656 − 0.753i)13-s + (−1.67 + 0.734i)14-s + (0.186 − 0.107i)15-s + 2.17·16-s − 0.758·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.968 - 0.250i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102881 + 0.808077i\)
\(L(\frac12)\) \(\approx\) \(0.102881 + 0.808077i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.06 - 2.42i)T \)
13 \( 1 + (-2.36 + 2.71i)T \)
good2 \( 1 - 2.58iT - 2T^{2} \)
3 \( 1 + (0.259 - 0.449i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.39 + 0.806i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.34 - 1.35i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.12T + 17T^{2} \)
19 \( 1 + (-3.18 + 1.84i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.98T + 23T^{2} \)
29 \( 1 + (-2.68 - 4.65i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-9.07 + 5.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.95iT - 37T^{2} \)
41 \( 1 + (6.66 - 3.85i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.67 - 2.90i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.913 - 0.527i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 11.4iT - 59T^{2} \)
61 \( 1 + (-1.46 - 2.53i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.7 - 6.79i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.17 - 0.675i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.88 - 4.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.69iT - 83T^{2} \)
89 \( 1 + 1.75iT - 89T^{2} \)
97 \( 1 + (13.4 + 7.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.97118413477645884851991268147, −13.86199443567006061363207153034, −12.82081517807083239645848257628, −11.49575939854236929926360378239, −9.768032453162149619297677914334, −8.547437872865687811885774012356, −7.908185700752904829056834577693, −6.55829517220568463113799243894, −5.27188710414152442926613317590, −4.34501875962581247912489375981, 1.26612256890437437309622037185, 3.52586268377497243789815218450, 4.33935097469234951997144780227, 6.71485993270863562220455265898, 8.360790708578977239248235477625, 9.603173921960597174735709023613, 10.63864451778223732063958098317, 11.66814975298971300007850876850, 11.98949995808972793448462448610, 13.53429236003749609406744102316

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