Properties

Label 2-91-1.1-c7-0-8
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  − 3.66·2-s + 42.3·3-s − 114.·4-s − 251.·5-s − 155.·6-s − 343·7-s + 888.·8-s − 393.·9-s + 921.·10-s − 2.92e3·11-s − 4.85e3·12-s − 2.19e3·13-s + 1.25e3·14-s − 1.06e4·15-s + 1.14e4·16-s + 2.77e4·17-s + 1.44e3·18-s + 3.43e4·19-s + 2.88e4·20-s − 1.45e4·21-s + 1.07e4·22-s − 2.16e4·23-s + 3.76e4·24-s − 1.47e4·25-s + 8.04e3·26-s − 1.09e5·27-s + 3.93e4·28-s + ⋯
L(s)  = 1  − 0.323·2-s + 0.905·3-s − 0.895·4-s − 0.900·5-s − 0.293·6-s − 0.377·7-s + 0.613·8-s − 0.180·9-s + 0.291·10-s − 0.662·11-s − 0.810·12-s − 0.277·13-s + 0.122·14-s − 0.815·15-s + 0.696·16-s + 1.37·17-s + 0.0583·18-s + 1.14·19-s + 0.806·20-s − 0.342·21-s + 0.214·22-s − 0.371·23-s + 0.555·24-s − 0.188·25-s + 0.0897·26-s − 1.06·27-s + 0.338·28-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

\(L(4)\) \(\approx\) \(1.184050826\)
\(L(\frac12)\) \(\approx\) \(1.184050826\)
\(L(\frac{9}{2})\) 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 \( 1 + 3.66T + 128T^{2} \)
3 \( 1 - 42.3T + 2.18e3T^{2} \)
5 \( 1 + 251.T + 7.81e4T^{2} \)
11 \( 1 + 2.92e3T + 1.94e7T^{2} \)
17 \( 1 - 2.77e4T + 4.10e8T^{2} \)
19 \( 1 - 3.43e4T + 8.93e8T^{2} \)
23 \( 1 + 2.16e4T + 3.40e9T^{2} \)
29 \( 1 - 1.15e5T + 1.72e10T^{2} \)
31 \( 1 - 1.30e4T + 2.75e10T^{2} \)
37 \( 1 - 4.81e5T + 9.49e10T^{2} \)
41 \( 1 - 3.16e5T + 1.94e11T^{2} \)
43 \( 1 - 7.51e4T + 2.71e11T^{2} \)
47 \( 1 - 5.00e5T + 5.06e11T^{2} \)
53 \( 1 + 4.66e5T + 1.17e12T^{2} \)
59 \( 1 - 1.34e6T + 2.48e12T^{2} \)
61 \( 1 - 2.22e6T + 3.14e12T^{2} \)
67 \( 1 + 2.36e6T + 6.06e12T^{2} \)
71 \( 1 - 6.19e5T + 9.09e12T^{2} \)
73 \( 1 - 4.50e6T + 1.10e13T^{2} \)
79 \( 1 + 1.88e5T + 1.92e13T^{2} \)
83 \( 1 + 5.10e6T + 2.71e13T^{2} \)
89 \( 1 - 1.60e6T + 4.42e13T^{2} \)
97 \( 1 + 8.11e5T + 8.07e13T^{2} \)
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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

−12.79855601663954017464238655390, −11.69301434076027506584294673142, −10.12667583731171861481096510045, −9.308793133193854476623161794553, −8.070774301789460284001701249687, −7.66859782573741137923503826714, −5.47437468325190947382748799713, −3.97886541521018069206200973171, −2.90862220437325154832540882806, −0.69464762461064572171608488659, 0.69464762461064572171608488659, 2.90862220437325154832540882806, 3.97886541521018069206200973171, 5.47437468325190947382748799713, 7.66859782573741137923503826714, 8.070774301789460284001701249687, 9.308793133193854476623161794553, 10.12667583731171861481096510045, 11.69301434076027506584294673142, 12.79855601663954017464238655390

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