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Properties

Label 2-91-91.9-c1-0-4
Degree $2$
Conductor $91$
Sign $0.245 + 0.969i$
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  + (−0.929 − 1.60i)2-s + 2.29·3-s + (−0.726 + 1.25i)4-s + (0.0986 − 0.170i)5-s + (−2.13 − 3.69i)6-s + (1.58 + 2.11i)7-s − 1.01·8-s + 2.26·9-s − 0.366·10-s − 4.18·11-s + (−1.66 + 2.88i)12-s + (−2.72 − 2.36i)13-s + (1.92 − 4.52i)14-s + (0.226 − 0.392i)15-s + (2.39 + 4.15i)16-s + (−0.420 + 0.728i)17-s + ⋯
L(s)  = 1  + (−0.656 − 1.13i)2-s + 1.32·3-s + (−0.363 + 0.629i)4-s + (0.0441 − 0.0764i)5-s + (−0.870 − 1.50i)6-s + (0.600 + 0.799i)7-s − 0.359·8-s + 0.754·9-s − 0.115·10-s − 1.26·11-s + (−0.481 + 0.833i)12-s + (−0.755 − 0.655i)13-s + (0.515 − 1.20i)14-s + (0.0584 − 0.101i)15-s + (0.599 + 1.03i)16-s + (−0.102 + 0.176i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.245 + 0.969i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796777 - 0.620452i\)
\(L(\frac12)\) \(\approx\) \(0.796777 - 0.620452i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.58 - 2.11i)T \)
13 \( 1 + (2.72 + 2.36i)T \)
good2 \( 1 + (0.929 + 1.60i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 - 2.29T + 3T^{2} \)
5 \( 1 + (-0.0986 + 0.170i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + 4.18T + 11T^{2} \)
17 \( 1 + (0.420 - 0.728i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 - 1.35T + 19T^{2} \)
23 \( 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.11 + 7.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.640 - 1.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.52 + 2.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.69 - 4.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.66 + 4.61i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.83 + 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.32 + 4.02i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.02 - 5.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + (-2.98 - 5.17i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (1.94 + 3.36i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.36 + 9.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.07T + 83T^{2} \)
89 \( 1 + (-5.99 - 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.73 + 16.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.70040376152775104902212578461, −12.70359104353909368191943058216, −11.64113649951937977592907181862, −10.44156795760381520574523725571, −9.485161721497378345205438791589, −8.547061795274441157485448485588, −7.74474203919757176662656348242, −5.34037698613389552495816846594, −3.11928989381995533090059555544, −2.19310979310703892056007369531, 2.81605100482040806701727653512, 4.86070233394752450882833618282, 6.83585824018019985783553279651, 7.74584760878747702375588522282, 8.439485287262024586254821923071, 9.500795899770856262630023202240, 10.68346563006418797618977416176, 12.44852944939270812391671704811, 13.88646519491788151213336670297, 14.38787747705352690859966665700

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