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Properties

Label 2-91-91.5-c1-0-4
Degree $2$
Conductor $91$
Sign $0.546 - 0.837i$
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  + (0.493 + 1.84i)2-s + (2.29 − 1.32i)3-s + (−1.41 + 0.818i)4-s + (−3.30 + 0.885i)5-s + (3.57 + 3.57i)6-s + (−1.12 − 2.39i)7-s + (0.489 + 0.489i)8-s + (2.00 − 3.47i)9-s + (−3.26 − 5.65i)10-s + (−0.445 + 1.66i)11-s + (−2.16 + 3.75i)12-s + (−3.57 − 0.501i)13-s + (3.84 − 3.26i)14-s + (−6.40 + 6.40i)15-s + (−2.29 + 3.97i)16-s + (−1.22 − 2.12i)17-s + ⋯
L(s)  = 1  + (0.349 + 1.30i)2-s + (1.32 − 0.764i)3-s + (−0.708 + 0.409i)4-s + (−1.47 + 0.396i)5-s + (1.45 + 1.45i)6-s + (−0.426 − 0.904i)7-s + (0.173 + 0.173i)8-s + (0.668 − 1.15i)9-s + (−1.03 − 1.78i)10-s + (−0.134 + 0.501i)11-s + (−0.625 + 1.08i)12-s + (−0.990 − 0.138i)13-s + (1.02 − 0.871i)14-s + (−1.65 + 1.65i)15-s + (−0.574 + 0.994i)16-s + (−0.297 − 0.515i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.546 - 0.837i$
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: \((2,\ 91,\ (\ :1/2),\ 0.546 - 0.837i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18379 + 0.640941i\)
\(L(\frac12)\) \(\approx\) \(1.18379 + 0.640941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (1.12 + 2.39i)T \)
13 \( 1 + (3.57 + 0.501i)T \)
good2 \( 1 + (-0.493 - 1.84i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (-2.29 + 1.32i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (3.30 - 0.885i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.445 - 1.66i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.03 + 1.34i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.97 - 2.29i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.184T + 29T^{2} \)
31 \( 1 + (-0.659 + 2.46i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.210 - 0.0563i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-4.63 - 4.63i)T + 41iT^{2} \)
43 \( 1 + 0.562iT - 43T^{2} \)
47 \( 1 + (-0.998 - 3.72i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.67 + 4.63i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (13.9 + 3.73i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.30 + 0.754i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.67 + 1.78i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-1.70 + 1.70i)T - 71iT^{2} \)
73 \( 1 + (-11.7 - 3.15i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.48 - 2.57i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.504 - 0.504i)T + 83iT^{2} \)
89 \( 1 + (1.92 + 7.20i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 + 12.0i)T + 97iT^{2} \)
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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

−14.38998430986431000109731360886, −13.62935883969318437624495595887, −12.61155714567242790403854157273, −11.19887634852888571125762866980, −9.433974833429634294664481030729, −7.918512047317146519169246999342, −7.45481658123511171670669020503, −6.90913095686274873607291680643, −4.60021116772692745835191388217, −3.13244812766741622200307368491, 2.75497805910701391969795803673, 3.62010497657673613431385103530, 4.76233094597182874255801735203, 7.56037986185605594570677883880, 8.717307833264235363653627181015, 9.550224762241696183379658834854, 10.79096257775186426178393531331, 11.96023079956548680272884950990, 12.55746337250409834388883079852, 13.82287565191154446700150135039

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