Properties

Label 2-91-1.1-c7-0-5
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 0.930·2-s + 23.4·3-s − 127.·4-s − 494.·5-s + 21.8·6-s + 343·7-s − 237.·8-s − 1.63e3·9-s − 460.·10-s − 2.42e3·11-s − 2.98e3·12-s + 2.19e3·13-s + 319.·14-s − 1.15e4·15-s + 1.60e4·16-s + 3.89e3·17-s − 1.52e3·18-s − 4.51e3·19-s + 6.28e4·20-s + 8.04e3·21-s − 2.25e3·22-s + 6.60e4·23-s − 5.56e3·24-s + 1.66e5·25-s + 2.04e3·26-s − 8.96e4·27-s − 4.36e4·28-s + ⋯
L(s)  = 1  + 0.0822·2-s + 0.501·3-s − 0.993·4-s − 1.76·5-s + 0.0412·6-s + 0.377·7-s − 0.163·8-s − 0.748·9-s − 0.145·10-s − 0.549·11-s − 0.497·12-s + 0.277·13-s + 0.0310·14-s − 0.887·15-s + 0.979·16-s + 0.192·17-s − 0.0615·18-s − 0.151·19-s + 1.75·20-s + 0.189·21-s − 0.0451·22-s + 1.13·23-s − 0.0822·24-s + 2.13·25-s + 0.0228·26-s − 0.876·27-s − 0.375·28-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

\(L(4)\) \(\approx\) \(0.9164833254\)
\(L(\frac12)\) \(\approx\) \(0.9164833254\)
\(L(\frac{9}{2})\) 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 \( 1 - 0.930T + 128T^{2} \)
3 \( 1 - 23.4T + 2.18e3T^{2} \)
5 \( 1 + 494.T + 7.81e4T^{2} \)
11 \( 1 + 2.42e3T + 1.94e7T^{2} \)
17 \( 1 - 3.89e3T + 4.10e8T^{2} \)
19 \( 1 + 4.51e3T + 8.93e8T^{2} \)
23 \( 1 - 6.60e4T + 3.40e9T^{2} \)
29 \( 1 - 6.15e3T + 1.72e10T^{2} \)
31 \( 1 - 1.76e4T + 2.75e10T^{2} \)
37 \( 1 + 5.16e5T + 9.49e10T^{2} \)
41 \( 1 - 4.70e5T + 1.94e11T^{2} \)
43 \( 1 + 1.07e4T + 2.71e11T^{2} \)
47 \( 1 - 3.26e5T + 5.06e11T^{2} \)
53 \( 1 - 1.91e6T + 1.17e12T^{2} \)
59 \( 1 - 5.58e5T + 2.48e12T^{2} \)
61 \( 1 + 1.32e6T + 3.14e12T^{2} \)
67 \( 1 - 2.71e6T + 6.06e12T^{2} \)
71 \( 1 + 2.18e6T + 9.09e12T^{2} \)
73 \( 1 + 2.12e6T + 1.10e13T^{2} \)
79 \( 1 + 3.62e6T + 1.92e13T^{2} \)
83 \( 1 - 8.69e6T + 2.71e13T^{2} \)
89 \( 1 + 2.94e6T + 4.42e13T^{2} \)
97 \( 1 + 1.35e7T + 8.07e13T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.67566278465273841686950802739, −11.68272479102877733741487033121, −10.64726690301683206820412054030, −8.916971766152688153155814180860, −8.331863080677190488459615756733, −7.40421743811334569922958995200, −5.29135701171280410491793095896, −4.10714538922132723131700884267, −3.10108183696087487386120079086, −0.57955221890404332003864515888, 0.57955221890404332003864515888, 3.10108183696087487386120079086, 4.10714538922132723131700884267, 5.29135701171280410491793095896, 7.40421743811334569922958995200, 8.331863080677190488459615756733, 8.916971766152688153155814180860, 10.64726690301683206820412054030, 11.68272479102877733741487033121, 12.67566278465273841686950802739

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