Properties

Label 2-91-91.54-c1-0-6
Degree $2$
Conductor $91$
Sign $-0.369 + 0.929i$
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  + (1.28 − 1.28i)2-s + (−2.65 − 1.53i)3-s − 1.30i·4-s + (1.07 − 0.288i)5-s + (−5.38 + 1.44i)6-s + (−0.978 − 2.45i)7-s + (0.899 + 0.899i)8-s + (3.21 + 5.56i)9-s + (1.01 − 1.75i)10-s + (0.124 − 0.0332i)11-s + (−1.99 + 3.45i)12-s + (3.59 + 0.291i)13-s + (−4.41 − 1.90i)14-s + (−3.30 − 0.884i)15-s + 4.90·16-s − 0.273·17-s + ⋯
L(s)  = 1  + (0.908 − 0.908i)2-s + (−1.53 − 0.886i)3-s − 0.650i·4-s + (0.480 − 0.128i)5-s + (−2.19 + 0.589i)6-s + (−0.369 − 0.929i)7-s + (0.317 + 0.317i)8-s + (1.07 + 1.85i)9-s + (0.319 − 0.553i)10-s + (0.0374 − 0.0100i)11-s + (−0.576 + 0.997i)12-s + (0.996 + 0.0807i)13-s + (−1.17 − 0.508i)14-s + (−0.852 − 0.228i)15-s + 1.22·16-s − 0.0663·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.369 + 0.929i$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :1/2),\ -0.369 + 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.598216 - 0.881438i\)
\(L(\frac12)\) \(\approx\) \(0.598216 - 0.881438i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.978 + 2.45i)T \)
13 \( 1 + (-3.59 - 0.291i)T \)
good2 \( 1 + (-1.28 + 1.28i)T - 2iT^{2} \)
3 \( 1 + (2.65 + 1.53i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.07 + 0.288i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.124 + 0.0332i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.273T + 17T^{2} \)
19 \( 1 + (1.02 - 3.83i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 0.491iT - 23T^{2} \)
29 \( 1 + (4.62 + 8.00i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.85 - 6.92i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 + (2.31 - 8.63i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-8.44 - 4.87i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.51 + 5.66i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.467 - 0.809i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 4.86i)T - 59iT^{2} \)
61 \( 1 + (-6.74 + 3.89i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.316 + 1.18i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.793 + 2.96i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-0.118 - 0.0318i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.72 - 9.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.07 + 8.07i)T + 83iT^{2} \)
89 \( 1 + (-2.99 + 2.99i)T - 89iT^{2} \)
97 \( 1 + (-2.39 + 0.641i)T + (84.0 - 48.5i)T^{2} \)
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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

−13.24528483423798768008905810802, −12.81772120653903552528501096838, −11.68619548305655582608573915381, −10.99890995513393708429242989462, −10.08346489801623670842740643305, −7.76692365001041201194458053001, −6.40512464420845354367786730515, −5.44227355148486047448044757220, −3.96931734486736805216553624280, −1.55242273357904087163689461817, 3.99068983527187667192597091184, 5.37694337279331595564383384257, 5.88869227315616747360650502528, 6.84866600797019625432923372201, 9.089175725530556560409572733438, 10.27727499101005155564501818332, 11.25530951064230577806938249700, 12.43016801701828901024262318633, 13.37052385633610508226339289241, 14.74851450397364429271436943976

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