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Properties

Label 2-91-7.4-c1-0-5
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: $\chi_{91} (53, \cdot )$
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

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

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