Properties

Label 2-91-91.47-c1-0-6
Degree $2$
Conductor $91$
Sign $0.954 + 0.298i$
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  + (2.38 − 0.639i)2-s + (−1.77 + 1.02i)3-s + (3.56 − 2.05i)4-s + (−0.423 − 1.58i)5-s + (−3.57 + 3.57i)6-s + (−0.716 + 2.54i)7-s + (3.69 − 3.69i)8-s + (0.589 − 1.02i)9-s + (−2.02 − 3.50i)10-s + (−5.55 − 1.48i)11-s + (−4.20 + 7.28i)12-s + (3.57 + 0.473i)13-s + (−0.0809 + 6.54i)14-s + (2.36 + 2.36i)15-s + (2.34 − 4.06i)16-s + (−0.991 − 1.71i)17-s + ⋯
L(s)  = 1  + (1.68 − 0.452i)2-s + (−1.02 + 0.590i)3-s + (1.78 − 1.02i)4-s + (−0.189 − 0.707i)5-s + (−1.45 + 1.45i)6-s + (−0.270 + 0.962i)7-s + (1.30 − 1.30i)8-s + (0.196 − 0.340i)9-s + (−0.640 − 1.10i)10-s + (−1.67 − 0.448i)11-s + (−1.21 + 2.10i)12-s + (0.991 + 0.131i)13-s + (−0.0216 + 1.74i)14-s + (0.611 + 0.611i)15-s + (0.587 − 1.01i)16-s + (−0.240 − 0.416i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.954 + 0.298i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.954 + 0.298i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.59716 - 0.243866i\)
\(L(\frac12)\) \(\approx\) \(1.59716 - 0.243866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.716 - 2.54i)T \)
13 \( 1 + (-3.57 - 0.473i)T \)
good2 \( 1 + (-2.38 + 0.639i)T + (1.73 - i)T^{2} \)
3 \( 1 + (1.77 - 1.02i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.423 + 1.58i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (5.55 + 1.48i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.991 + 1.71i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.246 + 0.918i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.06 - 1.77i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.83T + 29T^{2} \)
31 \( 1 + (-4.33 - 1.16i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.00 - 3.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (4.02 - 4.02i)T - 41iT^{2} \)
43 \( 1 + 5.30iT - 43T^{2} \)
47 \( 1 + (-0.448 + 0.120i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.31 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.10 - 11.5i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.38 + 2.52i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.57 - 5.87i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (4.84 + 4.84i)T + 71iT^{2} \)
73 \( 1 + (1.13 - 4.24i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.08 - 5.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-11.5 + 11.5i)T - 83iT^{2} \)
89 \( 1 + (3.51 - 0.941i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (7.09 - 7.09i)T - 97iT^{2} \)
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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

−13.62921860524308979933069770621, −12.97447018647313458717926106546, −11.97338718197978691686699669073, −11.23626372271520050984478858980, −10.33058302780649046805669431438, −8.519997991708159053651532539853, −6.28769984785755741221715153899, −5.32787611654820952519903431714, −4.73248922856310594121070521255, −2.92493677494817773271705926455, 3.16040769029032918882533824482, 4.71819545084503112497821865995, 5.99732189800184164993349772543, 6.80626811415126990216158269315, 7.72422030017291382273392220756, 10.60251668201812812109146037178, 11.11475014211334134505364666108, 12.47266777923065861292159262449, 13.06151973765944010084290929324, 13.90752422568451435411809193377

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