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Properties

Label 2-91-91.33-c1-0-5
Degree $2$
Conductor $91$
Sign $0.987 + 0.159i$
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  + (−0.320 + 1.19i)2-s − 2.22i·3-s + (0.401 + 0.231i)4-s + (−0.674 − 2.51i)5-s + (2.66 + 0.713i)6-s + (2.64 − 0.170i)7-s + (−2.15 + 2.15i)8-s − 1.95·9-s + 3.22·10-s + (−0.999 + 0.999i)11-s + (0.515 − 0.893i)12-s + (−0.445 + 3.57i)13-s + (−0.642 + 3.21i)14-s + (−5.60 + 1.50i)15-s + (−1.42 − 2.47i)16-s + (1.41 − 2.44i)17-s + ⋯
L(s)  = 1  + (−0.226 + 0.846i)2-s − 1.28i·3-s + (0.200 + 0.115i)4-s + (−0.301 − 1.12i)5-s + (1.08 + 0.291i)6-s + (0.997 − 0.0646i)7-s + (−0.763 + 0.763i)8-s − 0.650·9-s + 1.02·10-s + (−0.301 + 0.301i)11-s + (0.148 − 0.257i)12-s + (−0.123 + 0.992i)13-s + (−0.171 + 0.859i)14-s + (−1.44 + 0.387i)15-s + (−0.357 − 0.618i)16-s + (0.342 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.961252 - 0.0771079i\)
\(L(\frac12)\) \(\approx\) \(0.961252 - 0.0771079i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.64 + 0.170i)T \)
13 \( 1 + (0.445 - 3.57i)T \)
good2 \( 1 + (0.320 - 1.19i)T + (-1.73 - i)T^{2} \)
3 \( 1 + 2.22iT - 3T^{2} \)
5 \( 1 + (0.674 + 2.51i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.999 - 0.999i)T - 11iT^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.42 - 4.42i)T - 19iT^{2} \)
23 \( 1 + (-0.882 + 0.509i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.66 - 4.61i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (6.46 + 1.73i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-8.89 - 2.38i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.51 - 9.40i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.850 + 0.490i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.21 + 0.594i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.52 + 4.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.36 - 1.97i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 7.79iT - 61T^{2} \)
67 \( 1 + (9.24 + 9.24i)T + 67iT^{2} \)
71 \( 1 + (-2.97 + 11.0i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.26 + 8.43i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.78 - 4.82i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.445 + 0.445i)T - 83iT^{2} \)
89 \( 1 + (0.0396 - 0.147i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.87 - 0.771i)T + (84.0 + 48.5i)T^{2} \)
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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

−14.18597436724143842968268133622, −12.84072368205332442991208977469, −12.17649108103032175186192127746, −11.24750235752437058060872594688, −9.082713498970969351334241829830, −8.024352560088550979020632076497, −7.50213577608170640480123288783, −6.23000728344387800852529986882, −4.76204783357436630618670976074, −1.82189531788184256964631003223, 2.69503555945678813990979353663, 4.01897820423653832964321216491, 5.70426634851519864414704704171, 7.42908274599702584636567178522, 8.978665819949803638846910747361, 10.28781507913841766451829646000, 10.82881761161483313369508275718, 11.31883949127323044049086156920, 12.81957577005967696451825873339, 14.62806726776034173268531732479

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