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Properties

Label 2-91-13.9-c1-0-2
Degree $2$
Conductor $91$
Sign $0.583 - 0.811i$
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  + (0.355 − 0.615i)2-s + (−1.20 + 2.08i)3-s + (0.747 + 1.29i)4-s − 1.28·5-s + (0.855 + 1.48i)6-s + (0.5 + 0.866i)7-s + 2.48·8-s + (−1.39 − 2.42i)9-s + (−0.458 + 0.793i)10-s + (1.20 − 2.08i)11-s − 3.60·12-s + (1.25 − 3.38i)13-s + 0.710·14-s + (1.55 − 2.68i)15-s + (−0.613 + 1.06i)16-s + (−1.95 − 3.38i)17-s + ⋯
L(s)  = 1  + (0.251 − 0.434i)2-s + (−0.695 + 1.20i)3-s + (0.373 + 0.647i)4-s − 0.576·5-s + (0.349 + 0.604i)6-s + (0.188 + 0.327i)7-s + 0.877·8-s + (−0.466 − 0.807i)9-s + (−0.144 + 0.250i)10-s + (0.363 − 0.628i)11-s − 1.03·12-s + (0.347 − 0.937i)13-s + 0.189·14-s + (0.400 − 0.694i)15-s + (−0.153 + 0.265i)16-s + (−0.473 − 0.819i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.583 - 0.811i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.583 - 0.811i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.854637 + 0.438078i\)
\(L(\frac12)\) \(\approx\) \(0.854637 + 0.438078i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.25 + 3.38i)T \)
good2 \( 1 + (-0.355 + 0.615i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.20 - 2.08i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + 1.28T + 5T^{2} \)
11 \( 1 + (-1.20 + 2.08i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.94 - 5.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.16 + 5.47i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.80 - 4.86i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.20T + 31T^{2} \)
37 \( 1 + (-2.55 + 4.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.89 + 6.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.144 + 0.250i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.27T + 47T^{2} \)
53 \( 1 + 13.6T + 53T^{2} \)
59 \( 1 + (-2.01 - 3.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.30 - 3.98i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.78 - 6.56i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.61 + 6.25i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 15.0T + 73T^{2} \)
79 \( 1 + 9.30T + 79T^{2} \)
83 \( 1 + 1.36T + 83T^{2} \)
89 \( 1 + (-0.449 + 0.779i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.83 + 13.5i)T + (-48.5 + 84.0i)T^{2} \)
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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

−14.34682488027570077772998314399, −12.88587816488495172105977744887, −11.81817518642034313736296876105, −11.16023919221597904342252266492, −10.33226462264486833100974219959, −8.825428349477017618911683862160, −7.54794332249330086249902289740, −5.75126890898905878344211176071, −4.38592304884536978046225303563, −3.25585626399353990198167632791, 1.54151959001482983014460890972, 4.52032711300677936695871667532, 6.06677272383344985850022443475, 6.93921382332453102387667450538, 7.71857427659596176480219074764, 9.588854659042422940602621891710, 11.34290430013609946385398594773, 11.51088227671547309150968618632, 13.02596106157498602842496194692, 13.79657402858578961845453598371

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