Properties

Label 2-91-91.81-c1-0-5
Degree $2$
Conductor $91$
Sign $0.397 + 0.917i$
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  + (0.952 − 1.65i)2-s + 0.428·3-s + (−0.815 − 1.41i)4-s + (0.736 + 1.27i)5-s + (0.408 − 0.707i)6-s + (−2.62 − 0.311i)7-s + 0.702·8-s − 2.81·9-s + 2.80·10-s − 4.39·11-s + (−0.349 − 0.605i)12-s + (2.69 + 2.39i)13-s + (−3.01 + 4.03i)14-s + (0.315 + 0.546i)15-s + (2.30 − 3.98i)16-s + (0.601 + 1.04i)17-s + ⋯
L(s)  = 1  + (0.673 − 1.16i)2-s + 0.247·3-s + (−0.407 − 0.706i)4-s + (0.329 + 0.570i)5-s + (0.166 − 0.288i)6-s + (−0.993 − 0.117i)7-s + 0.248·8-s − 0.938·9-s + 0.887·10-s − 1.32·11-s + (−0.100 − 0.174i)12-s + (0.748 + 0.663i)13-s + (−0.806 + 1.07i)14-s + (0.0814 + 0.141i)15-s + (0.575 − 0.996i)16-s + (0.145 + 0.252i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.397 + 0.917i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.397 + 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13238 - 0.743718i\)
\(L(\frac12)\) \(\approx\) \(1.13238 - 0.743718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.62 + 0.311i)T \)
13 \( 1 + (-2.69 - 2.39i)T \)
good2 \( 1 + (-0.952 + 1.65i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 - 0.428T + 3T^{2} \)
5 \( 1 + (-0.736 - 1.27i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 4.39T + 11T^{2} \)
17 \( 1 + (-0.601 - 1.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 3.24T + 19T^{2} \)
23 \( 1 + (-2.21 + 3.84i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0837 + 0.145i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.62 - 4.54i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.52 - 6.10i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.58 + 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.0113 - 0.0197i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.84 + 10.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.0708 + 0.122i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.67 - 4.62i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 11.5T + 61T^{2} \)
67 \( 1 - 4.13T + 67T^{2} \)
71 \( 1 + (-4.98 + 8.63i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.62 - 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.387 + 0.670i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 16.0T + 83T^{2} \)
89 \( 1 + (3.27 - 5.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.74 - 3.02i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.64686853981771858663908290673, −12.93260300649249535544746771650, −11.79262756332152520627545143453, −10.72215415470714070835250365320, −10.03031337306835370899532674560, −8.520121825407119655694018246346, −6.84186418381644143599937427204, −5.34955969592241799584942325025, −3.49160821166836662000830581435, −2.57410502882543459336409725158, 3.24472151618467011588974359923, 5.28910117630892829738891696677, 5.85092102902327517521043883033, 7.37443527480713649089968317023, 8.447001911959619430498884380515, 9.702624843808053253425592211203, 11.10306832220830201724432802115, 12.93504544791923987631954489742, 13.23920140194018215877307343033, 14.29520029511420702331498968190

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