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Properties

Label 2-91-91.73-c1-0-7
Degree $2$
Conductor $91$
Sign $-0.736 + 0.676i$
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  + (0.697 − 2.60i)2-s + (−0.657 − 0.379i)3-s + (−4.55 − 2.63i)4-s + (2.44 + 0.654i)5-s + (−1.44 + 1.44i)6-s + (0.722 + 2.54i)7-s + (−6.22 + 6.22i)8-s + (−1.21 − 2.09i)9-s + (3.40 − 5.89i)10-s + (0.557 + 2.08i)11-s + (1.99 + 3.46i)12-s + (1.44 − 3.30i)13-s + (7.13 − 0.104i)14-s + (−1.35 − 1.35i)15-s + (6.59 + 11.4i)16-s + (−0.700 + 1.21i)17-s + ⋯
L(s)  = 1  + (0.493 − 1.84i)2-s + (−0.379 − 0.219i)3-s + (−2.27 − 1.31i)4-s + (1.09 + 0.292i)5-s + (−0.590 + 0.590i)6-s + (0.272 + 0.962i)7-s + (−2.19 + 2.19i)8-s + (−0.403 − 0.699i)9-s + (1.07 − 1.86i)10-s + (0.168 + 0.627i)11-s + (0.576 + 0.999i)12-s + (0.401 − 0.915i)13-s + (1.90 − 0.0279i)14-s + (−0.350 − 0.350i)15-s + (1.64 + 2.85i)16-s + (−0.169 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.736 + 0.676i$
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: \((2,\ 91,\ (\ :1/2),\ -0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.403536 - 1.03511i\)
\(L(\frac12)\) \(\approx\) \(0.403536 - 1.03511i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.722 - 2.54i)T \)
13 \( 1 + (-1.44 + 3.30i)T \)
good2 \( 1 + (-0.697 + 2.60i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (0.657 + 0.379i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.44 - 0.654i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.557 - 2.08i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.700 - 1.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.02 - 0.541i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.13 - 0.657i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.56T + 29T^{2} \)
31 \( 1 + (-1.88 - 7.03i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (2.20 + 0.591i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.69 - 2.69i)T - 41iT^{2} \)
43 \( 1 - 0.437iT - 43T^{2} \)
47 \( 1 + (-2.07 + 7.74i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.26 + 2.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.54 - 2.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.57 + 3.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.548 - 0.146i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (10.7 + 10.7i)T + 71iT^{2} \)
73 \( 1 + (-11.8 + 3.18i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.19 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.82 - 3.82i)T - 83iT^{2} \)
89 \( 1 + (0.0134 - 0.0501i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.43 + 9.43i)T - 97iT^{2} \)
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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

−13.31994575679568661250826493062, −12.41670257671853248384265362309, −11.72826487049608319702879122531, −10.61727610383850638930111686060, −9.697314864961827487914861568887, −8.762449686652853377844994541298, −6.05926579385159472463900297182, −5.20325465682554084234481959536, −3.23799400439605023315972129755, −1.81360002597548942441751327401, 4.20443928610951397827436490278, 5.37378280566419494178313222708, 6.23717071150100729297651057530, 7.47104113843213983276167008585, 8.673308995244327554260578538939, 9.768960279893395507981450186932, 11.34487718917321181245728257112, 13.14716042535009461526072822363, 13.85370908072830038395254388680, 14.19290833092873278029906047009

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