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Properties

Label 2-91-91.34-c1-0-4
Degree $2$
Conductor $91$
Sign $-0.0404 + 0.999i$
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  + (−0.403 − 0.403i)2-s − 1.23i·3-s − 1.67i·4-s + (−1.03 + 1.03i)5-s + (−0.496 + 0.496i)6-s + (−0.450 − 2.60i)7-s + (−1.48 + 1.48i)8-s + 1.48·9-s + 0.832·10-s + (−0.596 + 0.596i)11-s − 2.06·12-s + (3.59 − 0.296i)13-s + (−0.869 + 1.23i)14-s + (1.27 + 1.27i)15-s − 2.15·16-s + 7.34·17-s + ⋯
L(s)  = 1  + (−0.284 − 0.284i)2-s − 0.711i·3-s − 0.837i·4-s + (−0.461 + 0.461i)5-s + (−0.202 + 0.202i)6-s + (−0.170 − 0.985i)7-s + (−0.523 + 0.523i)8-s + 0.493·9-s + 0.263·10-s + (−0.179 + 0.179i)11-s − 0.595·12-s + (0.996 − 0.0822i)13-s + (−0.232 + 0.329i)14-s + (0.328 + 0.328i)15-s − 0.539·16-s + 1.78·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0404 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.0404 + 0.999i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.0404 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.569193 - 0.592706i\)
\(L(\frac12)\) \(\approx\) \(0.569193 - 0.592706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.450 + 2.60i)T \)
13 \( 1 + (-3.59 + 0.296i)T \)
good2 \( 1 + (0.403 + 0.403i)T + 2iT^{2} \)
3 \( 1 + 1.23iT - 3T^{2} \)
5 \( 1 + (1.03 - 1.03i)T - 5iT^{2} \)
11 \( 1 + (0.596 - 0.596i)T - 11iT^{2} \)
17 \( 1 - 7.34T + 17T^{2} \)
19 \( 1 + (3.59 - 3.59i)T - 19iT^{2} \)
23 \( 1 - 4.44iT - 23T^{2} \)
29 \( 1 + 3.54T + 29T^{2} \)
31 \( 1 + (-1.27 + 1.27i)T - 31iT^{2} \)
37 \( 1 + (-2.88 + 2.88i)T - 37iT^{2} \)
41 \( 1 + (1.23 - 1.23i)T - 41iT^{2} \)
43 \( 1 + 8.66iT - 43T^{2} \)
47 \( 1 + (2.52 + 2.52i)T + 47iT^{2} \)
53 \( 1 + 9.79T + 53T^{2} \)
59 \( 1 + (-1.08 - 1.08i)T + 59iT^{2} \)
61 \( 1 - 7.10iT - 61T^{2} \)
67 \( 1 + (-8.76 - 8.76i)T + 67iT^{2} \)
71 \( 1 + (1.46 + 1.46i)T + 71iT^{2} \)
73 \( 1 + (-0.103 - 0.103i)T + 73iT^{2} \)
79 \( 1 + 4.79T + 79T^{2} \)
83 \( 1 + (12.3 - 12.3i)T - 83iT^{2} \)
89 \( 1 + (-6.89 - 6.89i)T + 89iT^{2} \)
97 \( 1 + (-6.05 + 6.05i)T - 97iT^{2} \)
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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

−13.80656201581319437762562493091, −12.80702502089739829318656502181, −11.52619594744873661072292701285, −10.52676816448769140893626338981, −9.762866300969025477500442055171, −8.006552062181290814110916885732, −7.04972455114949885526030315069, −5.77865797319384529196238754067, −3.74843316107361357390718761227, −1.37484451771622661573916242012, 3.28769654531385555764249880728, 4.63693595906412472972589541507, 6.32070001225240707976573494996, 7.943489602344617178572803335396, 8.733918210508951539607890245369, 9.761639133629122835726963167293, 11.20932187701196969890719734673, 12.37324841396081890422494091866, 12.97760650438473411900102572614, 14.68715345438894706095325912671

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