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Properties

Label 2-91-91.5-c1-0-7
Degree $2$
Conductor $91$
Sign $-0.521 - 0.853i$
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  + (−0.639 − 2.38i)2-s + (−1.77 + 1.02i)3-s + (−3.56 + 2.05i)4-s + (−1.58 + 0.423i)5-s + (3.57 + 3.57i)6-s + (−2.54 − 0.716i)7-s + (3.69 + 3.69i)8-s + (0.589 − 1.02i)9-s + (2.02 + 3.50i)10-s + (1.48 − 5.55i)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  + (−0.452 − 1.68i)2-s + (−1.02 + 0.590i)3-s + (−1.78 + 1.02i)4-s + (−0.707 + 0.189i)5-s + (1.45 + 1.45i)6-s + (−0.962 − 0.270i)7-s + (1.30 + 1.30i)8-s + (0.196 − 0.340i)9-s + (0.640 + 1.10i)10-s + (0.448 − 1.67i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.521 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.521 - 0.853i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.521 - 0.853i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0513369 + 0.0915200i\)
\(L(\frac12)\) \(\approx\) \(0.0513369 + 0.0915200i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−12.85971082229152802789554094032, −11.84118615904983768286031138025, −11.26689582116989277193622342654, −10.38326186754805423729485458828, −9.558611095265756133307526283966, −8.172831787050814690695403190068, −6.13981008693127561889060710116, −4.30511378081091072410206491226, −3.18564411095796655286323075734, −0.16168386649015083634404578518, 4.63920892333123282818905888844, 5.89957986254053027795593433397, 6.96236947704667478709491529809, 7.51526878917342754137782646027, 9.105097156257343329878293227382, 10.07025172286154170827686939402, 12.12220722910987782821277006845, 12.41879322230566733072650608888, 14.05651581552118129448109046489, 15.24631650589810310981810718385

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