Properties

Label 2-91-1.1-c7-0-33
Degree $2$
Conductor $91$
Sign $1$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 19.9·2-s + 56.3·3-s + 270.·4-s + 1.53·5-s + 1.12e3·6-s + 343·7-s + 2.84e3·8-s + 984.·9-s + 30.5·10-s + 702.·11-s + 1.52e4·12-s + 2.19e3·13-s + 6.84e3·14-s + 86.1·15-s + 2.21e4·16-s − 1.36e4·17-s + 1.96e4·18-s + 9.52e3·19-s + 413.·20-s + 1.93e4·21-s + 1.40e4·22-s + 6.87e4·23-s + 1.60e5·24-s − 7.81e4·25-s + 4.38e4·26-s − 6.77e4·27-s + 9.27e4·28-s + ⋯
L(s)  = 1  + 1.76·2-s + 1.20·3-s + 2.11·4-s + 0.00547·5-s + 2.12·6-s + 0.377·7-s + 1.96·8-s + 0.450·9-s + 0.00966·10-s + 0.159·11-s + 2.54·12-s + 0.277·13-s + 0.666·14-s + 0.00659·15-s + 1.35·16-s − 0.673·17-s + 0.794·18-s + 0.318·19-s + 0.0115·20-s + 0.455·21-s + 0.280·22-s + 1.17·23-s + 2.36·24-s − 0.999·25-s + 0.489·26-s − 0.662·27-s + 0.798·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $1$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(8.429901019\)
\(L(\frac12)\) \(\approx\) \(8.429901019\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - 343T \)
13 \( 1 - 2.19e3T \)
good2 \( 1 - 19.9T + 128T^{2} \)
3 \( 1 - 56.3T + 2.18e3T^{2} \)
5 \( 1 - 1.53T + 7.81e4T^{2} \)
11 \( 1 - 702.T + 1.94e7T^{2} \)
17 \( 1 + 1.36e4T + 4.10e8T^{2} \)
19 \( 1 - 9.52e3T + 8.93e8T^{2} \)
23 \( 1 - 6.87e4T + 3.40e9T^{2} \)
29 \( 1 + 8.42e4T + 1.72e10T^{2} \)
31 \( 1 - 7.84e4T + 2.75e10T^{2} \)
37 \( 1 + 1.72e5T + 9.49e10T^{2} \)
41 \( 1 + 8.18e4T + 1.94e11T^{2} \)
43 \( 1 - 2.38e5T + 2.71e11T^{2} \)
47 \( 1 + 8.91e5T + 5.06e11T^{2} \)
53 \( 1 + 5.55e5T + 1.17e12T^{2} \)
59 \( 1 - 2.18e6T + 2.48e12T^{2} \)
61 \( 1 + 1.75e5T + 3.14e12T^{2} \)
67 \( 1 + 1.17e6T + 6.06e12T^{2} \)
71 \( 1 + 3.77e5T + 9.09e12T^{2} \)
73 \( 1 - 7.57e5T + 1.10e13T^{2} \)
79 \( 1 - 3.59e6T + 1.92e13T^{2} \)
83 \( 1 - 8.17e6T + 2.71e13T^{2} \)
89 \( 1 - 4.92e6T + 4.42e13T^{2} \)
97 \( 1 + 3.38e6T + 8.07e13T^{2} \)
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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

−13.19174521898087199689741214694, −11.89437302487455603860484725828, −10.96331972677873169831347136723, −9.260706243731600531667536884510, −7.968996386376129932003694027255, −6.72945673559640612410704415883, −5.33642194396455064757164785528, −4.06316704120972146876320170989, −3.05642196884090982834010073390, −1.90825323555893724232474652023, 1.90825323555893724232474652023, 3.05642196884090982834010073390, 4.06316704120972146876320170989, 5.33642194396455064757164785528, 6.72945673559640612410704415883, 7.968996386376129932003694027255, 9.260706243731600531667536884510, 10.96331972677873169831347136723, 11.89437302487455603860484725828, 13.19174521898087199689741214694

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