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Properties

Label 2-91-91.74-c1-0-6
Degree $2$
Conductor $91$
Sign $0.800 + 0.599i$
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  + 1.85·2-s + (−1.14 − 1.98i)3-s + 1.45·4-s + (0.0986 + 0.170i)5-s + (−2.13 − 3.69i)6-s + (1.03 + 2.43i)7-s − 1.01·8-s + (−1.13 + 1.95i)9-s + (0.183 + 0.317i)10-s + (2.09 + 3.62i)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 − 4.79·16-s + 0.841·17-s + ⋯
L(s)  = 1  + 1.31·2-s + (−0.662 − 1.14i)3-s + 0.726·4-s + (0.0441 + 0.0764i)5-s + (−0.870 − 1.50i)6-s + (0.392 + 0.919i)7-s − 0.359·8-s + (−0.377 + 0.653i)9-s + (0.0579 + 0.100i)10-s + (0.630 + 1.09i)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 − 1.19·16-s + 0.204·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.800 + 0.599i$
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: \((2,\ 91,\ (\ :1/2),\ 0.800 + 0.599i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37996 - 0.459951i\)
\(L(\frac12)\) \(\approx\) \(1.37996 - 0.459951i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.03 - 2.43i)T \)
13 \( 1 + (2.72 + 2.36i)T \)
good2 \( 1 - 1.85T + 2T^{2} \)
3 \( 1 + (1.14 + 1.98i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.0986 - 0.170i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.09 - 3.62i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.841T + 17T^{2} \)
19 \( 1 + (0.675 - 1.17i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.11T + 23T^{2} \)
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 \( 1 - 3.04T + 37T^{2} \)
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 \( 1 - 6.05T + 59T^{2} \)
61 \( 1 + (-5.68 + 9.84i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.69 + 11.6i)T + (-33.5 + 58.0i)T^{2} \)
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 \( 1 + 11.9T + 89T^{2} \)
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

−13.88327755460646201352246929871, −12.53221974378779041913080373462, −12.35757035998622164543234194097, −11.53800946709852407986562000789, −9.722561495905344493101263395449, −7.994121223195895754578933878066, −6.62234041539874342954531203810, −5.77219128723676646835073294721, −4.53259423803456685867788893299, −2.35076883251026167736757838044, 3.60551067540227648957971749912, 4.55531960880949055981187932399, 5.49069616491588983198232385358, 6.85812154078220756942460857945, 8.870910729969378693686349720464, 10.18473673684928846982986341723, 11.22516323999060025817111608453, 11.96660447844270333470153651002, 13.37968043599532539891469608578, 14.23713713503722120656371099440

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