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Properties

Label 2-91-7.4-c1-0-2
Degree $2$
Conductor $91$
Sign $-0.361 - 0.932i$
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  + (1.00 + 1.73i)2-s + (−0.879 + 1.52i)3-s + (−1.01 + 1.76i)4-s + (−0.452 − 0.784i)5-s − 3.53·6-s + (0.237 − 2.63i)7-s − 0.0686·8-s + (−0.0471 − 0.0816i)9-s + (0.909 − 1.57i)10-s + (−0.358 + 0.620i)11-s + (−1.78 − 3.09i)12-s + 13-s + (4.82 − 2.23i)14-s + 1.59·15-s + (1.96 + 3.40i)16-s + (−1.17 + 2.03i)17-s + ⋯
L(s)  = 1  + (0.710 + 1.22i)2-s + (−0.507 + 0.879i)3-s + (−0.508 + 0.880i)4-s + (−0.202 − 0.350i)5-s − 1.44·6-s + (0.0898 − 0.995i)7-s − 0.0242·8-s + (−0.0157 − 0.0272i)9-s + (0.287 − 0.498i)10-s + (−0.107 + 0.187i)11-s + (−0.516 − 0.894i)12-s + 0.277·13-s + (1.28 − 0.596i)14-s + 0.411·15-s + (0.491 + 0.850i)16-s + (−0.285 + 0.494i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.361 - 0.932i$
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: \((2,\ 91,\ (\ :1/2),\ -0.361 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.672443 + 0.982338i\)
\(L(\frac12)\) \(\approx\) \(0.672443 + 0.982338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.237 + 2.63i)T \)
13 \( 1 - T \)
good2 \( 1 + (-1.00 - 1.73i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (0.879 - 1.52i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.452 + 0.784i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.358 - 0.620i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.17 - 2.03i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.31 + 5.74i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.87 + 3.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.25T + 29T^{2} \)
31 \( 1 + (0.785 - 1.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.60 + 4.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.92T + 41T^{2} \)
43 \( 1 + 9.43T + 43T^{2} \)
47 \( 1 + (-4.15 - 7.20i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.04 - 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.358 - 0.620i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.82 - 10.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.69 - 8.13i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + (-1.73 + 3.00i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.50 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.54T + 83T^{2} \)
89 \( 1 + (6.02 + 10.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.43T + 97T^{2} \)
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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

−14.60392738192361276560433824903, −13.60195387985080456643250353259, −12.68927234435081492920885263221, −10.97525115350448860444764131721, −10.31761220347988093153010821796, −8.611472007712786356739646341112, −7.34952472058576369051691349193, −6.20917471725666159071687681210, −4.74040104744342984958208378829, −4.24429525564441193119084043889, 1.89202839583541384462979097762, 3.51892770271674247484985365349, 5.31618080681637123914031186301, 6.59877797205958214651940996322, 8.103002994600222945331782699948, 9.764835585924940091771901797573, 11.08989564986948798115249852632, 11.80458218704909615986894830037, 12.50615895860765964737112535821, 13.31393303437901177366850286674

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