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Properties

Label 2-91-7.2-c1-0-2
Degree $2$
Conductor $91$
Sign $0.782 - 0.622i$
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  + (0.0978 − 0.169i)2-s + (0.129 + 0.224i)3-s + (0.980 + 1.69i)4-s + (−1.96 + 3.40i)5-s + 0.0508·6-s + (1.12 − 2.39i)7-s + 0.775·8-s + (1.46 − 2.53i)9-s + (0.384 + 0.666i)10-s + (−2.25 − 3.90i)11-s + (−0.254 + 0.441i)12-s + 13-s + (−0.296 − 0.424i)14-s − 1.02·15-s + (−1.88 + 3.26i)16-s + (1.14 + 1.97i)17-s + ⋯
L(s)  = 1  + (0.0691 − 0.119i)2-s + (0.0749 + 0.129i)3-s + (0.490 + 0.849i)4-s + (−0.879 + 1.52i)5-s + 0.0207·6-s + (0.424 − 0.905i)7-s + 0.274·8-s + (0.488 − 0.846i)9-s + (0.121 + 0.210i)10-s + (−0.679 − 1.17i)11-s + (−0.0735 + 0.127i)12-s + 0.277·13-s + (−0.0791 − 0.113i)14-s − 0.263·15-s + (−0.471 + 0.816i)16-s + (0.276 + 0.479i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.782 - 0.622i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.782 - 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.990930 + 0.346137i\)
\(L(\frac12)\) \(\approx\) \(0.990930 + 0.346137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.12 + 2.39i)T \)
13 \( 1 - T \)
good2 \( 1 + (-0.0978 + 0.169i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.129 - 0.224i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.96 - 3.40i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.25 + 3.90i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.14 - 1.97i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.893 + 1.54i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.870 - 1.50i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.65T + 29T^{2} \)
31 \( 1 + (2.80 + 4.85i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 - 6.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 - 6.81T + 43T^{2} \)
47 \( 1 + (1.77 - 3.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.64 + 2.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.25 + 3.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.77 - 6.53i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.33 - 10.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.54T + 71T^{2} \)
73 \( 1 + (0.540 + 0.935i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.395 - 0.685i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.14T + 83T^{2} \)
89 \( 1 + (-5.63 + 9.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.81T + 97T^{2} \)
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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

−14.24894172403577753301398978367, −13.18419569858679450362537620407, −11.77105742490266202143406960035, −11.08961684713626655067702046245, −10.29877513222967572502085995892, −8.258727870116567356059780712649, −7.42001519043501803986358006198, −6.48913649677470728983897143890, −3.91118410282086055755930759134, −3.15985279368455953602078928041, 1.82135772864216292595207452466, 4.73310608718834228244856945712, 5.34784786711109292399120711796, 7.33145278620487392651276192409, 8.293142850145652991984629791580, 9.532351880538946842153940906932, 10.81846023474111740827462525551, 12.08911783345295843275719286669, 12.66445228355075830939777076700, 14.03363150054773480310264145027

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