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Properties

Label 2-91-91.16-c1-0-1
Degree $2$
Conductor $91$
Sign $0.969 - 0.245i$
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  − 1.90·2-s + (−0.214 + 0.371i)3-s + 1.63·4-s + (0.736 − 1.27i)5-s + (0.408 − 0.707i)6-s + (1.58 + 2.11i)7-s + 0.702·8-s + (1.40 + 2.43i)9-s + (−1.40 + 2.43i)10-s + (2.19 − 3.80i)11-s + (−0.349 + 0.605i)12-s + (2.69 + 2.39i)13-s + (−3.01 − 4.03i)14-s + (0.315 + 0.546i)15-s − 4.60·16-s − 1.20·17-s + ⋯
L(s)  = 1  − 1.34·2-s + (−0.123 + 0.214i)3-s + 0.815·4-s + (0.329 − 0.570i)5-s + (0.166 − 0.288i)6-s + (0.598 + 0.801i)7-s + 0.248·8-s + (0.469 + 0.813i)9-s + (−0.443 + 0.768i)10-s + (0.662 − 1.14i)11-s + (−0.100 + 0.174i)12-s + (0.748 + 0.663i)13-s + (−0.806 − 1.07i)14-s + (0.0814 + 0.141i)15-s − 1.15·16-s − 0.291·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.969 - 0.245i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.969 - 0.245i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.576458 + 0.0717422i\)
\(L(\frac12)\) \(\approx\) \(0.576458 + 0.0717422i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.58 - 2.11i)T \)
13 \( 1 + (-2.69 - 2.39i)T \)
good2 \( 1 + 1.90T + 2T^{2} \)
3 \( 1 + (0.214 - 0.371i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.736 + 1.27i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.19 + 3.80i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
19 \( 1 + (1.62 + 2.80i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.43T + 23T^{2} \)
29 \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.62 + 4.54i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 7.05T + 37T^{2} \)
41 \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.84 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0708 - 0.122i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.34T + 59T^{2} \)
61 \( 1 + (5.77 + 9.99i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.06 - 3.58i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.62 + 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.387 - 0.670i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 - 6.55T + 89T^{2} \)
97 \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.06847494399047882914994779317, −13.13320571208889022710442854228, −11.46430224234113307168011959196, −10.87888075105104611806449727922, −9.407300660985387741089847723931, −8.799277999682287982094250276349, −7.84636563311050374250092532007, −6.12268935248963430456661375642, −4.57519299751439993358271528029, −1.72563343605294342870054241683, 1.51908533655827400352476466536, 4.18697088988592255677401499061, 6.47607079543751022548417620216, 7.36875551612933266975547147177, 8.512679179293615699492156229588, 9.864657149160381039803818026450, 10.39694015094341403308773024683, 11.58214372436119730036680721030, 12.92277519425206551263718891857, 14.19312166818781083484072408101

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