Properties

Label 2-91-91.30-c1-0-2
Degree $2$
Conductor $91$
Sign $0.721 - 0.692i$
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  + (−1.19 + 0.689i)2-s + 2.88·3-s + (−0.0491 + 0.0850i)4-s + (0.697 + 0.402i)5-s + (−3.44 + 1.98i)6-s + (−2.25 − 1.38i)7-s − 2.89i·8-s + 5.30·9-s − 1.11·10-s + 5.27i·11-s + (−0.141 + 0.245i)12-s + (−2.36 − 2.72i)13-s + (3.64 + 0.0965i)14-s + (2.01 + 1.16i)15-s + (1.89 + 3.28i)16-s + (−0.280 + 0.485i)17-s + ⋯
L(s)  = 1  + (−0.844 + 0.487i)2-s + 1.66·3-s + (−0.0245 + 0.0425i)4-s + (0.312 + 0.180i)5-s + (−1.40 + 0.811i)6-s + (−0.852 − 0.522i)7-s − 1.02i·8-s + 1.76·9-s − 0.351·10-s + 1.58i·11-s + (−0.0408 + 0.0707i)12-s + (−0.656 − 0.754i)13-s + (0.974 + 0.0257i)14-s + (0.519 + 0.299i)15-s + (0.474 + 0.821i)16-s + (−0.0679 + 0.117i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.721 - 0.692i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.721 - 0.692i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899027 + 0.361930i\)
\(L(\frac12)\) \(\approx\) \(0.899027 + 0.361930i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.25 + 1.38i)T \)
13 \( 1 + (2.36 + 2.72i)T \)
good2 \( 1 + (1.19 - 0.689i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 2.88T + 3T^{2} \)
5 \( 1 + (-0.697 - 0.402i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 5.27iT - 11T^{2} \)
17 \( 1 + (0.280 - 0.485i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.84iT - 19T^{2} \)
23 \( 1 + (0.802 + 1.38i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.14 - 1.97i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.01 - 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.07 + 0.620i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.803 - 0.463i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.22 - 3.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.32 - 1.92i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.72 + 4.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-9.52 - 5.49i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 7.30T + 61T^{2} \)
67 \( 1 - 7.34iT - 67T^{2} \)
71 \( 1 + (8.06 - 4.65i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.33 - 2.50i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.68 - 9.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.81iT - 83T^{2} \)
89 \( 1 + (-4.33 + 2.50i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.22 + 5.32i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.39376235608943346015776759033, −13.13738033841919726172253879248, −12.69120458002639311025565219100, −10.11702347920084170078704078497, −9.694101776428212904681638827301, −8.719635460370607410356265497230, −7.50643305777776288226821095762, −6.94880114191295320247126462180, −4.17750125001286238247246978802, −2.66913973246867474274493636693, 2.11018146007775990517914738098, 3.49333552070925410516908475152, 5.81505727663046856578769443209, 7.75273805733257294743310673664, 8.830109510603256205986711020625, 9.319083431138138339150765836214, 10.20302524334830401900439769567, 11.71988585365725020299438954691, 13.18612007698662043226565233544, 13.99958529328319564310660296243

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