Properties

Label 2-91-91.89-c1-0-3
Degree $2$
Conductor $91$
Sign $0.325 - 0.945i$
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  + (1.74 + 1.74i)2-s + (−0.146 − 0.0844i)3-s + 4.08i·4-s + (−0.638 − 2.38i)5-s + (−0.107 − 0.402i)6-s + (−2.04 − 1.67i)7-s + (−3.63 + 3.63i)8-s + (−1.48 − 2.57i)9-s + (3.04 − 5.26i)10-s + (1.17 + 4.40i)11-s + (0.344 − 0.596i)12-s + (1.54 + 3.25i)13-s + (−0.640 − 6.49i)14-s + (−0.107 + 0.402i)15-s − 4.49·16-s + 0.112·17-s + ⋯
L(s)  = 1  + (1.23 + 1.23i)2-s + (−0.0844 − 0.0487i)3-s + 2.04i·4-s + (−0.285 − 1.06i)5-s + (−0.0440 − 0.164i)6-s + (−0.773 − 0.634i)7-s + (−1.28 + 1.28i)8-s + (−0.495 − 0.857i)9-s + (0.962 − 1.66i)10-s + (0.355 + 1.32i)11-s + (0.0994 − 0.172i)12-s + (0.429 + 0.903i)13-s + (−0.171 − 1.73i)14-s + (−0.0278 + 0.103i)15-s − 1.12·16-s + 0.0273·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.325 - 0.945i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.325 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24013 + 0.884811i\)
\(L(\frac12)\) \(\approx\) \(1.24013 + 0.884811i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.04 + 1.67i)T \)
13 \( 1 + (-1.54 - 3.25i)T \)
good2 \( 1 + (-1.74 - 1.74i)T + 2iT^{2} \)
3 \( 1 + (0.146 + 0.0844i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.638 + 2.38i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.17 - 4.40i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 - 0.112T + 17T^{2} \)
19 \( 1 + (3.32 + 0.891i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 - 0.652iT - 23T^{2} \)
29 \( 1 + (2.82 + 4.88i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.34 - 1.43i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-1.09 + 1.09i)T - 37iT^{2} \)
41 \( 1 + (10.6 + 2.86i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-6.08 - 3.51i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.72 - 1.53i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.41 + 4.17i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.79 - 2.79i)T + 59iT^{2} \)
61 \( 1 + (-13.2 + 7.66i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.07 - 1.62i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.56 + 0.955i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.651 - 2.43i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-6.11 + 10.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.34 - 3.34i)T - 83iT^{2} \)
89 \( 1 + (6.14 + 6.14i)T + 89iT^{2} \)
97 \( 1 + (-1.04 - 3.89i)T + (-84.0 + 48.5i)T^{2} \)
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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

−14.37110805020415828253256389263, −13.28381964086441143663240306234, −12.56723236732752103445383343017, −11.77250329954259375189334617857, −9.617992736698745327083318665187, −8.446877237637058421737678336886, −7.02021399405527572144349949009, −6.25964544240291773366556117736, −4.69117283384596957172864029017, −3.84355491842482306686551159384, 2.72613552877741723007278778823, 3.53656872174332723145150463628, 5.43715917573163551098025645172, 6.37819032829722527172484476677, 8.455967881300241862480214477499, 10.21982025017912325952295050494, 10.94679663959130546364973292834, 11.65538219483916773584511719347, 12.87589988862216279868353410521, 13.67867986016500884980550928481

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