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Properties

Label 2-91-91.45-c1-0-2
Degree $2$
Conductor $91$
Sign $-0.354 - 0.935i$
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  + (−1.56 + 1.56i)2-s + (0.959 − 0.554i)3-s − 2.91i·4-s + (−0.784 + 2.92i)5-s + (−0.635 + 2.37i)6-s + (2.25 + 1.38i)7-s + (1.43 + 1.43i)8-s + (−0.885 + 1.53i)9-s + (−3.36 − 5.82i)10-s + (−0.188 + 0.705i)11-s + (−1.61 − 2.79i)12-s + (−2.65 − 2.44i)13-s + (−5.70 + 1.35i)14-s + (0.869 + 3.24i)15-s + 1.33·16-s + 3.24·17-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)2-s + (0.554 − 0.319i)3-s − 1.45i·4-s + (−0.351 + 1.31i)5-s + (−0.259 + 0.968i)6-s + (0.851 + 0.524i)7-s + (0.506 + 0.506i)8-s + (−0.295 + 0.511i)9-s + (−1.06 − 1.84i)10-s + (−0.0569 + 0.212i)11-s + (−0.466 − 0.807i)12-s + (−0.735 − 0.678i)13-s + (−1.52 + 0.362i)14-s + (0.224 + 0.838i)15-s + 0.334·16-s + 0.787·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.354 - 0.935i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.354 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.377627 + 0.546801i\)
\(L(\frac12)\) \(\approx\) \(0.377627 + 0.546801i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.65 + 2.44i)T \)
good2 \( 1 + (1.56 - 1.56i)T - 2iT^{2} \)
3 \( 1 + (-0.959 + 0.554i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.784 - 2.92i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.188 - 0.705i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 - 3.24T + 17T^{2} \)
19 \( 1 + (-3.85 + 1.03i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + 5.96iT - 23T^{2} \)
29 \( 1 + (-2.78 + 4.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.99 + 1.07i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (6.97 + 6.97i)T + 37iT^{2} \)
41 \( 1 + (-2.46 + 0.660i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.73 - 3.30i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.69 + 0.454i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.37 + 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.33 - 2.33i)T - 59iT^{2} \)
61 \( 1 + (-6.30 - 3.63i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.27 + 1.68i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.06 + 2.16i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-1.62 - 6.08i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-7.87 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.4 - 10.4i)T + 83iT^{2} \)
89 \( 1 + (6.62 - 6.62i)T - 89iT^{2} \)
97 \( 1 + (-2.37 + 8.86i)T + (-84.0 - 48.5i)T^{2} \)
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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

−14.73283107850076296428063656500, −14.01009029374612744291481502234, −12.16590340559139750242918996263, −10.84292594200163328322978527147, −9.862509687675275755722419991983, −8.397795324852311420580065270300, −7.77369820727973054947662327386, −6.93658703410624879967836749906, −5.42578558005704123264897462880, −2.71793152196696642917200198242, 1.29322077906759351010725357167, 3.44126747444579305594485016560, 5.00022921287505198807018787888, 7.68911017647718475845204913265, 8.567893286182326293526326407863, 9.333512926769634203702813117600, 10.28338455399436174416501858375, 11.91011827735640965024909737017, 11.93457508576109474279807311891, 13.66084529427158847594262030822

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