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Properties

Label 2-91-91.31-c1-0-5
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: $\chi_{91} (31, \cdot )$
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

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

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