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Properties

Label 2-91-13.3-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.979 + 0.202i$
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  + (−0.866 − 1.5i)2-s + (−1.36 − 2.36i)3-s + (−0.5 + 0.866i)4-s + 1.73·5-s + (−2.36 + 4.09i)6-s + (−0.5 + 0.866i)7-s − 1.73·8-s + (−2.23 + 3.86i)9-s + (−1.49 − 2.59i)10-s + (−0.633 − 1.09i)11-s + 2.73·12-s + (3.59 + 0.232i)13-s + 1.73·14-s + (−2.36 − 4.09i)15-s + (2.49 + 4.33i)16-s + (3.86 − 6.69i)17-s + ⋯
L(s)  = 1  + (−0.612 − 1.06i)2-s + (−0.788 − 1.36i)3-s + (−0.250 + 0.433i)4-s + 0.774·5-s + (−0.965 + 1.67i)6-s + (−0.188 + 0.327i)7-s − 0.612·8-s + (−0.744 + 1.28i)9-s + (−0.474 − 0.821i)10-s + (−0.191 − 0.331i)11-s + 0.788·12-s + (0.997 + 0.0643i)13-s + 0.462·14-s + (−0.610 − 1.05i)15-s + (0.624 + 1.08i)16-s + (0.937 − 1.62i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.202i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.979 + 0.202i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.979 + 0.202i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0634710 - 0.620695i\)
\(L(\frac12)\) \(\approx\) \(0.0634710 - 0.620695i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-3.59 - 0.232i)T \)
good2 \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.73T + 5T^{2} \)
11 \( 1 + (0.633 + 1.09i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-3.86 + 6.69i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.36 + 4.09i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.19T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.59 - 4.5i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.0980 + 0.169i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 9.92T + 53T^{2} \)
59 \( 1 + (3.63 - 6.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.40 + 4.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.09 - 5.36i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 - 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.19T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + 2.19T + 83T^{2} \)
89 \( 1 + (6.46 + 11.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.19 - 5.53i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.24586092230948158654969932437, −12.23005857609586705230924641082, −11.60417710998818345196829919010, −10.52807817338731503102535610609, −9.427670507004874679649652828066, −8.090634438141046613290676055716, −6.46336141615095059662223825229, −5.69536135995479125566446277246, −2.64588382454827270263349499521, −1.12534378376491676230474080064, 3.88086092302963128809729415964, 5.64231538960605410821936467083, 6.21385964025188172171290512662, 7.924442681759987638474619549944, 9.253876937203867667498890562736, 10.04946838483038308866525894690, 10.93689010970627290765972950814, 12.37968902497733386537962002930, 13.88196892102361452841898463818, 15.15666101548764240559220355731

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