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Properties

Label 2-91-91.25-c1-0-4
Degree $2$
Conductor $91$
Sign $0.922 + 0.385i$
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  + (−0.929 + 0.536i)2-s + (1.21 − 2.10i)3-s + (−0.424 + 0.734i)4-s + (0.541 − 0.312i)5-s + 2.60i·6-s + (2.34 − 1.21i)7-s − 3.05i·8-s + (−1.45 − 2.52i)9-s + (−0.335 + 0.581i)10-s + (−0.613 − 0.354i)11-s + (1.03 + 1.78i)12-s + (0.848 + 3.50i)13-s + (−1.53 + 2.39i)14-s − 1.52i·15-s + (0.791 + 1.37i)16-s + (−1.67 + 2.89i)17-s + ⋯
L(s)  = 1  + (−0.657 + 0.379i)2-s + (0.701 − 1.21i)3-s + (−0.212 + 0.367i)4-s + (0.242 − 0.139i)5-s + 1.06i·6-s + (0.888 − 0.459i)7-s − 1.08i·8-s + (−0.485 − 0.840i)9-s + (−0.106 + 0.183i)10-s + (−0.185 − 0.106i)11-s + (0.297 + 0.515i)12-s + (0.235 + 0.971i)13-s + (−0.409 + 0.638i)14-s − 0.392i·15-s + (0.197 + 0.342i)16-s + (−0.405 + 0.702i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.922 + 0.385i$
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: \((2,\ 91,\ (\ :1/2),\ 0.922 + 0.385i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866463 - 0.173663i\)
\(L(\frac12)\) \(\approx\) \(0.866463 - 0.173663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.34 + 1.21i)T \)
13 \( 1 + (-0.848 - 3.50i)T \)
good2 \( 1 + (0.929 - 0.536i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-1.21 + 2.10i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.541 + 0.312i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.613 + 0.354i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.50 - 2.60i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.21 + 3.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.59T + 29T^{2} \)
31 \( 1 + (-3.80 - 2.19i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.366 - 0.211i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.01iT - 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + (6.99 - 4.03i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.348 + 0.603i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.54 - 4.93i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.34 + 4.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.02 + 5.21i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 + (4.40 + 2.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.95 - 3.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2iT - 83T^{2} \)
89 \( 1 + (-11.5 + 6.68i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.202iT - 97T^{2} \)
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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

−13.87038517321903808416074222408, −13.12096848693716332986123776688, −12.19915783578700963354780321028, −10.70008035524334243250847834858, −9.103612252720291020051980708767, −8.275406811112421405413525971683, −7.53449086104323600136500656970, −6.42966156960714116187380308346, −4.11818090296131782691924359164, −1.81181873650907394438582298479, 2.45617401562077076141043778453, 4.44151668219954097927642427719, 5.62108612676731222687781827847, 8.023765587431898354649048384878, 8.894154837952367796641388971325, 9.775155368452811041179035812224, 10.62321265690933101193849556298, 11.51212239808598100438667527446, 13.35847632260112077411351582578, 14.48161824863592306715297304665

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