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Properties

Label 2-91-91.73-c1-0-1
Degree $2$
Conductor $91$
Sign $-0.0794 - 0.996i$
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  + (−0.411 + 1.53i)2-s + (0.436 + 0.251i)3-s + (−0.454 − 0.262i)4-s + (−0.0769 − 0.0206i)5-s + (−0.565 + 0.565i)6-s + (1.52 + 2.16i)7-s + (−1.65 + 1.65i)8-s + (−1.37 − 2.37i)9-s + (0.0632 − 0.109i)10-s + (−1.09 − 4.08i)11-s + (−0.132 − 0.229i)12-s + (0.565 + 3.56i)13-s + (−3.94 + 1.44i)14-s + (−0.0283 − 0.0283i)15-s + (−2.38 − 4.13i)16-s + (2.90 − 5.02i)17-s + ⋯
L(s)  = 1  + (−0.290 + 1.08i)2-s + (0.251 + 0.145i)3-s + (−0.227 − 0.131i)4-s + (−0.0343 − 0.00921i)5-s + (−0.231 + 0.231i)6-s + (0.575 + 0.818i)7-s + (−0.585 + 0.585i)8-s + (−0.457 − 0.792i)9-s + (0.0200 − 0.0346i)10-s + (−0.329 − 1.23i)11-s + (−0.0381 − 0.0661i)12-s + (0.156 + 0.987i)13-s + (−1.05 + 0.386i)14-s + (−0.00732 − 0.00732i)15-s + (−0.596 − 1.03i)16-s + (0.704 − 1.21i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0794 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0794 - 0.996i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0794 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651793 + 0.705810i\)
\(L(\frac12)\) \(\approx\) \(0.651793 + 0.705810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.52 - 2.16i)T \)
13 \( 1 + (-0.565 - 3.56i)T \)
good2 \( 1 + (0.411 - 1.53i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (-0.436 - 0.251i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.0769 + 0.0206i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.09 + 4.08i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.90 + 5.02i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.11 - 1.36i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-0.755 + 0.436i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.362T + 29T^{2} \)
31 \( 1 + (0.361 + 1.34i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.76 + 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.70 - 7.70i)T - 41iT^{2} \)
43 \( 1 + 2.65iT - 43T^{2} \)
47 \( 1 + (0.748 - 2.79i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (5.26 - 9.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 - 0.573i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.63 + 2.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.76 + 2.61i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.65 - 3.65i)T + 71iT^{2} \)
73 \( 1 + (11.4 - 3.08i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.91 + 4.91i)T - 83iT^{2} \)
89 \( 1 + (2.08 - 7.78i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.04 + 6.04i)T - 97iT^{2} \)
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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

−14.40442886261472096045371082142, −13.94485669627495186024248670442, −11.87398276494008106178511228781, −11.53976667586312851598472405955, −9.479830049699507156837885064576, −8.662300012206158071130623979989, −7.72905548000276973768380440831, −6.28919236492594071673921105301, −5.32746070293469737556200367457, −3.04849702837512113497116076119, 1.76655867246288545538674683083, 3.46049243010264297972879501103, 5.28242477552219900653723370998, 7.26505505797785620717111958528, 8.260076256086244284886243719256, 9.907292844504787269480857267879, 10.52620267753647985647077730553, 11.51600675558234210968212670830, 12.63469854090890163762480870943, 13.55451223609448032030900220963

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