Properties

Label 2-91-91.74-c1-0-4
Degree $2$
Conductor $91$
Sign $0.982 - 0.188i$
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  + 1.55·2-s + (0.244 + 0.423i)3-s + 0.417·4-s + (0.595 + 1.03i)5-s + (0.380 + 0.658i)6-s + (−2.44 − 1.01i)7-s − 2.46·8-s + (1.38 − 2.39i)9-s + (0.926 + 1.60i)10-s + (−1.05 − 1.83i)11-s + (0.102 + 0.176i)12-s + (2.86 + 2.19i)13-s + (−3.79 − 1.58i)14-s + (−0.291 + 0.504i)15-s − 4.66·16-s − 0.906·17-s + ⋯
L(s)  = 1  + 1.09·2-s + (0.141 + 0.244i)3-s + 0.208·4-s + (0.266 + 0.461i)5-s + (0.155 + 0.268i)6-s + (−0.922 − 0.385i)7-s − 0.870·8-s + (0.460 − 0.796i)9-s + (0.292 + 0.507i)10-s + (−0.319 − 0.552i)11-s + (0.0294 + 0.0510i)12-s + (0.793 + 0.608i)13-s + (−1.01 − 0.423i)14-s + (−0.0752 + 0.130i)15-s − 1.16·16-s − 0.219·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.982 - 0.188i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.982 - 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48303 + 0.140838i\)
\(L(\frac12)\) \(\approx\) \(1.48303 + 0.140838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.44 + 1.01i)T \)
13 \( 1 + (-2.86 - 2.19i)T \)
good2 \( 1 - 1.55T + 2T^{2} \)
3 \( 1 + (-0.244 - 0.423i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.595 - 1.03i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.05 + 1.83i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 0.906T + 17T^{2} \)
19 \( 1 + (3.34 - 5.79i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 3.59T + 23T^{2} \)
29 \( 1 + (4.25 - 7.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.64 + 4.57i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
41 \( 1 + (0.768 - 1.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.71 + 4.70i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.59 - 2.75i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.41 + 2.44i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 10.2T + 59T^{2} \)
61 \( 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.26 - 2.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.86 + 4.96i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.03 + 5.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (3.10 + 5.37i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.07883479931745162391450408983, −13.13661580884291146824312754955, −12.43005967859820526229938118632, −10.99964022217774494544446391386, −9.821564504447169770855584111387, −8.751211304553328590149226388240, −6.72573767094484853686076855630, −5.96743247130696181165604926837, −4.16169913094958830987836304505, −3.22984496892227500874324299375, 2.75702149941998175409532461712, 4.47578615373994744792370653383, 5.60397517133561060689009966916, 6.87434185488195645711013659510, 8.559995177069430509793327840878, 9.624762272587006696403208114707, 11.05781898383187491507186549384, 12.56291395049297522078005595185, 13.14829077539790113206673103600, 13.54117812416410673344257849813

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