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Properties

Label 2-91-91.16-c1-0-0
Degree $2$
Conductor $91$
Sign $-0.601 - 0.798i$
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  − 0.851·2-s + (−0.330 + 0.572i)3-s − 1.27·4-s + (−1.72 + 2.98i)5-s + (0.281 − 0.487i)6-s + (−2.57 − 0.617i)7-s + 2.78·8-s + (1.28 + 2.21i)9-s + (1.46 − 2.53i)10-s + (0.448 − 0.777i)11-s + (0.421 − 0.730i)12-s + (−3.07 + 1.88i)13-s + (2.18 + 0.525i)14-s + (−1.13 − 1.97i)15-s + 0.178·16-s + 1.93·17-s + ⋯
L(s)  = 1  − 0.601·2-s + (−0.190 + 0.330i)3-s − 0.637·4-s + (−0.769 + 1.33i)5-s + (0.114 − 0.198i)6-s + (−0.972 − 0.233i)7-s + 0.985·8-s + (0.427 + 0.739i)9-s + (0.463 − 0.802i)10-s + (0.135 − 0.234i)11-s + (0.121 − 0.210i)12-s + (−0.852 + 0.522i)13-s + (0.585 + 0.140i)14-s + (−0.293 − 0.508i)15-s + 0.0445·16-s + 0.469·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.601 - 0.798i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.601 - 0.798i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.177451 + 0.355736i\)
\(L(\frac12)\) \(\approx\) \(0.177451 + 0.355736i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.57 + 0.617i)T \)
13 \( 1 + (3.07 - 1.88i)T \)
good2 \( 1 + 0.851T + 2T^{2} \)
3 \( 1 + (0.330 - 0.572i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.72 - 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.448 + 0.777i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 - 1.93T + 17T^{2} \)
19 \( 1 + (0.519 + 0.898i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + (-0.917 - 1.58i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.56 - 7.91i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 + (-2.66 - 4.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.95 + 3.39i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (3.59 - 6.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.69 - 8.12i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.510T + 59T^{2} \)
61 \( 1 + (0.718 + 1.24i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.22 + 7.31i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.72 + 2.98i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.45 + 9.44i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.04 + 10.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.51T + 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (0.253 - 0.438i)T + (-48.5 - 84.0i)T^{2} \)
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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

−14.45353848006101049863513965181, −13.56658421476350821492391401083, −12.24790121391126754396280051165, −10.77455024135556811906807050382, −10.28620499940829679447630133924, −9.141028365110287160088974532982, −7.62411435973076939288279091632, −6.81900865071141253252545193097, −4.74818788486658011413846152620, −3.29798952854065220458022770473, 0.63091211390540319228885632549, 3.93957760638755693920728497407, 5.28070665609173707208846684148, 7.10834918617921672930999908620, 8.288156026789267064231410485179, 9.283226296690955590543801030425, 10.02774198793735375386511027296, 11.94474046935467690108321170515, 12.63750239703854009605067957248, 13.27359634478475569688698350235

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