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Properties

Label 2-91-91.30-c1-0-1
Degree $2$
Conductor $91$
Sign $0.357 - 0.934i$
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  + (−2.24 + 1.29i)2-s + 0.518·3-s + (2.35 − 4.07i)4-s + (1.39 + 0.806i)5-s + (−1.16 + 0.671i)6-s + (2.62 + 0.292i)7-s + 6.99i·8-s − 2.73·9-s − 4.17·10-s + 2.70i·11-s + (1.21 − 2.11i)12-s + (2.36 + 2.71i)13-s + (−6.27 + 2.74i)14-s + (0.723 + 0.417i)15-s + (−4.34 − 7.53i)16-s + (1.56 − 2.70i)17-s + ⋯
L(s)  = 1  + (−1.58 + 0.915i)2-s + 0.299·3-s + (1.17 − 2.03i)4-s + (0.624 + 0.360i)5-s + (−0.474 + 0.273i)6-s + (0.993 + 0.110i)7-s + 2.47i·8-s − 0.910·9-s − 1.31·10-s + 0.815i·11-s + (0.351 − 0.609i)12-s + (0.656 + 0.753i)13-s + (−1.67 + 0.734i)14-s + (0.186 + 0.107i)15-s + (−1.08 − 1.88i)16-s + (0.379 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.357 - 0.934i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.502905 + 0.346169i\)
\(L(\frac12)\) \(\approx\) \(0.502905 + 0.346169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 0.292i)T \)
13 \( 1 + (-2.36 - 2.71i)T \)
good2 \( 1 + (2.24 - 1.29i)T + (1 - 1.73i)T^{2} \)
3 \( 1 - 0.518T + 3T^{2} \)
5 \( 1 + (-1.39 - 0.806i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 - 2.70iT - 11T^{2} \)
17 \( 1 + (-1.56 + 2.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 3.68iT - 19T^{2} \)
23 \( 1 + (-0.993 - 1.71i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (9.07 - 5.23i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.15 + 2.97i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.66 + 3.85i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.67 + 2.90i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.913 + 0.527i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.63 + 6.29i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.89 + 5.71i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + 2.92T + 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 + (-1.17 + 0.675i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.88 + 4.55i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.10 + 5.37i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.69iT - 83T^{2} \)
89 \( 1 + (-1.52 + 0.879i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.4 - 7.74i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.53744321647206253168099911594, −13.86152452170692073243963851808, −11.63652150555816668028631304187, −10.75300085246752616567214718151, −9.524271193523454164091624503221, −8.780059039943379313694487188530, −7.71094559610854377897642858816, −6.61656006320986357874818321604, −5.32189199970069830593701397090, −2.02107103678672107090617869672, 1.53246087765476644687847980795, 3.25868719150855701582783645432, 5.79059902803887089591529323438, 7.921762754392104985549229072533, 8.430425270332240229290708156010, 9.415653729281807459623017974201, 10.70027456234586883884022351875, 11.25514537889700153941836666830, 12.49966855978683482536896769996, 13.71855645330016722263708647946

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