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Properties

Label 2-91-91.80-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.988 + 0.153i$
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  + (−0.629 − 2.35i)2-s − 1.97i·3-s + (−3.39 + 1.96i)4-s + (0.0608 − 0.227i)5-s + (−4.63 + 1.24i)6-s + (2.26 + 1.37i)7-s + (3.30 + 3.30i)8-s − 0.884·9-s − 0.572·10-s + (−2.02 − 2.02i)11-s + (3.86 + 6.69i)12-s + (2.03 + 2.97i)13-s + (1.79 − 6.18i)14-s + (−0.447 − 0.119i)15-s + (1.76 − 3.05i)16-s + (−2.01 − 3.49i)17-s + ⋯
L(s)  = 1  + (−0.445 − 1.66i)2-s − 1.13i·3-s + (−1.69 + 0.980i)4-s + (0.0272 − 0.101i)5-s + (−1.89 + 0.506i)6-s + (0.855 + 0.518i)7-s + (1.16 + 1.16i)8-s − 0.294·9-s − 0.180·10-s + (−0.609 − 0.609i)11-s + (1.11 + 1.93i)12-s + (0.563 + 0.826i)13-s + (0.480 − 1.65i)14-s + (−0.115 − 0.0309i)15-s + (0.441 − 0.764i)16-s + (−0.488 − 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 + 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.988 + 0.153i$
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: \((2,\ 91,\ (\ :1/2),\ -0.988 + 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0604214 - 0.780056i\)
\(L(\frac12)\) \(\approx\) \(0.0604214 - 0.780056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.26 - 1.37i)T \)
13 \( 1 + (-2.03 - 2.97i)T \)
good2 \( 1 + (0.629 + 2.35i)T + (-1.73 + i)T^{2} \)
3 \( 1 + 1.97iT - 3T^{2} \)
5 \( 1 + (-0.0608 + 0.227i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (2.02 + 2.02i)T + 11iT^{2} \)
17 \( 1 + (2.01 + 3.49i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.88 + 3.88i)T + 19iT^{2} \)
23 \( 1 + (-5.23 - 3.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.22 + 0.595i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (10.0 - 2.68i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.0713 + 0.266i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.91 - 2.25i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.14 + 0.842i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.96 - 6.87i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.702 + 0.188i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 2.28iT - 61T^{2} \)
67 \( 1 + (-1.88 + 1.88i)T - 67iT^{2} \)
71 \( 1 + (0.362 + 1.35i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-3.50 - 13.0i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.91 + 3.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.19 + 4.19i)T + 83iT^{2} \)
89 \( 1 + (0.893 + 3.33i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (2.61 - 0.701i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.21667276176854510717702226141, −12.38556517597778194518981090089, −11.39055368745541769721097686167, −10.81880111727381078497330154701, −9.085450789206123484409642628656, −8.441097744535395892631779830879, −6.90246818556380574779220691654, −4.82566832890024185669456825885, −2.75759791472285238116119294638, −1.39068347601616318954163042678, 4.27815244880589343026688142193, 5.21575743400652102150885229788, 6.64752994884842608036739770117, 7.990166384016533521222057212429, 8.729141981493108747388686993850, 10.21837811206742149773821541309, 10.68678463072833299213475824261, 12.84496620237156832383723181335, 14.17335304111903822037548498349, 15.06381633461751180463345830902

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