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Properties

Label 2-91-91.25-c1-0-6
Degree $2$
Conductor $91$
Sign $0.705 + 0.708i$
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  + (1.84 − 1.06i)2-s + (0.0894 − 0.154i)3-s + (1.25 − 2.18i)4-s + (−3.12 + 1.80i)5-s − 0.380i·6-s + (1.20 − 2.35i)7-s − 1.10i·8-s + (1.48 + 2.57i)9-s + (−3.83 + 6.63i)10-s + (−3.45 − 1.99i)11-s + (−0.225 − 0.389i)12-s + (−2.51 + 2.58i)13-s + (−0.274 − 5.61i)14-s + 0.644i·15-s + (1.34 + 2.33i)16-s + (2.39 − 4.14i)17-s + ⋯
L(s)  = 1  + (1.30 − 0.751i)2-s + (0.0516 − 0.0894i)3-s + (0.629 − 1.09i)4-s + (−1.39 + 0.806i)5-s − 0.155i·6-s + (0.457 − 0.889i)7-s − 0.389i·8-s + (0.494 + 0.856i)9-s + (−1.21 + 2.09i)10-s + (−1.04 − 0.601i)11-s + (−0.0649 − 0.112i)12-s + (−0.698 + 0.715i)13-s + (−0.0734 − 1.50i)14-s + 0.166i·15-s + (0.337 + 0.583i)16-s + (0.580 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.705 + 0.708i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43834 - 0.597538i\)
\(L(\frac12)\) \(\approx\) \(1.43834 - 0.597538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.20 + 2.35i)T \)
13 \( 1 + (2.51 - 2.58i)T \)
good2 \( 1 + (-1.84 + 1.06i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-0.0894 + 0.154i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.12 - 1.80i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.45 + 1.99i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.39 + 4.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.72 + 1.57i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.08 + 1.88i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.57T + 29T^{2} \)
31 \( 1 + (-1.28 - 0.743i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.29 + 2.48i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.11iT - 41T^{2} \)
43 \( 1 + 1.43T + 43T^{2} \)
47 \( 1 + (-0.882 + 0.509i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.01 - 5.22i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.24 - 2.45i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.01 - 1.76i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.38 - 1.95i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.80iT - 71T^{2} \)
73 \( 1 + (2.67 + 1.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.984 + 1.70i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.66iT - 83T^{2} \)
89 \( 1 + (11.0 - 6.39i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.35iT - 97T^{2} \)
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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

−13.91505535563894903927743691524, −12.98038635740211318925080215176, −11.69009827368317257830932612380, −11.17119480062452862142933634920, −10.27689672482422588857151659376, −7.88140958993088559784534581589, −7.23434025951219482207093075815, −5.08308445155651164629507007054, −4.08370806588513418339395173203, −2.76968858853606132719351905843, 3.53591857254118058678515257632, 4.73258963835571134295601281304, 5.66117176873066564320897433799, 7.45039223983452915496532238754, 8.130174203011639473463283674592, 9.781402721082455622122784177543, 11.67249332570376680317889879115, 12.53508577079952963044055548030, 12.83015662967116697647928488026, 14.62725014029239243917909923761

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