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Properties

Label 2-91-91.80-c1-0-5
Degree $2$
Conductor $91$
Sign $0.192 + 0.981i$
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  + (−0.127 − 0.474i)2-s − 2.44i·3-s + (1.52 − 0.879i)4-s + (−0.931 + 3.47i)5-s + (−1.15 + 0.310i)6-s + (−0.668 − 2.55i)7-s + (−1.30 − 1.30i)8-s − 2.98·9-s + 1.76·10-s + (1.49 + 1.49i)11-s + (−2.15 − 3.72i)12-s + (−0.582 + 3.55i)13-s + (−1.12 + 0.642i)14-s + (8.50 + 2.27i)15-s + (1.30 − 2.26i)16-s + (0.572 + 0.990i)17-s + ⋯
L(s)  = 1  + (−0.0898 − 0.335i)2-s − 1.41i·3-s + (0.761 − 0.439i)4-s + (−0.416 + 1.55i)5-s + (−0.473 + 0.126i)6-s + (−0.252 − 0.967i)7-s + (−0.461 − 0.461i)8-s − 0.993·9-s + 0.558·10-s + (0.451 + 0.451i)11-s + (−0.620 − 1.07i)12-s + (−0.161 + 0.986i)13-s + (−0.301 + 0.171i)14-s + (2.19 + 0.588i)15-s + (0.326 − 0.565i)16-s + (0.138 + 0.240i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.192 + 0.981i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.192 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.780874 - 0.642321i\)
\(L(\frac12)\) \(\approx\) \(0.780874 - 0.642321i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.668 + 2.55i)T \)
13 \( 1 + (0.582 - 3.55i)T \)
good2 \( 1 + (0.127 + 0.474i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 2.44iT - 3T^{2} \)
5 \( 1 + (0.931 - 3.47i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.49 - 1.49i)T + 11iT^{2} \)
17 \( 1 + (-0.572 - 0.990i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.22 - 3.22i)T + 19iT^{2} \)
23 \( 1 + (1.28 + 0.741i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.75 - 4.77i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.85 - 1.56i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.61 + 0.700i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.21 + 4.54i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.55 + 2.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.51 + 1.74i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.74 - 3.02i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (11.0 + 2.95i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 9.46iT - 61T^{2} \)
67 \( 1 + (-5.44 + 5.44i)T - 67iT^{2} \)
71 \( 1 + (-1.74 - 6.52i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.46 - 5.46i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.91 - 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.06 + 5.06i)T + 83iT^{2} \)
89 \( 1 + (2.58 + 9.63i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-13.1 + 3.52i)T + (84.0 - 48.5i)T^{2} \)
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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

−14.16321148461214454280273645693, −12.57645758891935572971539720752, −11.69432999294527206585409770068, −10.84422278408568893993890744934, −9.842387853369165405515953138797, −7.60361630540947487572780149232, −6.95779806714110695907921289671, −6.38895556896584307615248933414, −3.43189117594849023254460183494, −1.80488072357391552630536107090, 3.26468136247556651590896565052, 4.83116964514757918004185302066, 5.85175985689679688359297214783, 7.924345095921918711957951639103, 8.861117118403807108731113633641, 9.675523267292317897081472172541, 11.27553806817273835108944405636, 12.04178840550894064736648591885, 13.06394332861337014984946237328, 14.93990173811776281217175419150

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