↑

Properties

Label 2-91-13.12-c7-0-39
Degree $2$
Conductor $91$
Sign $-0.562 - 0.826i$
Analytic cond. $28.4270$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12.6i·2-s − 83.1·3-s − 31.0·4-s + 6.92i·5-s + 1.04e3i·6-s − 343i·7-s − 1.22e3i·8-s + 4.72e3·9-s + 87.3·10-s − 7.24e3i·11-s + 2.58e3·12-s + (6.55e3 − 4.45e3i)13-s − 4.32e3·14-s − 575. i·15-s − 1.93e4·16-s − 2.15e4·17-s + ⋯
L(s)  = 1  − 1.11i·2-s − 1.77·3-s − 0.242·4-s + 0.0247i·5-s + 1.98i·6-s − 0.377i·7-s − 0.844i·8-s + 2.15·9-s + 0.0276·10-s − 1.64i·11-s + 0.431·12-s + (0.826 − 0.562i)13-s − 0.421·14-s − 0.0440i·15-s − 1.18·16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.562 - 0.826i$
Analytic conductor: \(28.4270\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{91} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 91,\ (\ :7/2),\ -0.562 - 0.826i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.386501 + 0.730202i\)
\(L(\frac12)\) \(\approx\) \(0.386501 + 0.730202i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + 343iT \)
13 \( 1 + (-6.55e3 + 4.45e3i)T \)
good2 \( 1 + 12.6iT - 128T^{2} \)
3 \( 1 + 83.1T + 2.18e3T^{2} \)
5 \( 1 - 6.92iT - 7.81e4T^{2} \)
11 \( 1 + 7.24e3iT - 1.94e7T^{2} \)
17 \( 1 + 2.15e4T + 4.10e8T^{2} \)
19 \( 1 + 2.96e4iT - 8.93e8T^{2} \)
23 \( 1 - 1.03e5T + 3.40e9T^{2} \)
29 \( 1 + 5.48e4T + 1.72e10T^{2} \)
31 \( 1 + 8.10e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.08e5iT - 9.49e10T^{2} \)
41 \( 1 - 3.47e5iT - 1.94e11T^{2} \)
43 \( 1 + 8.89e5T + 2.71e11T^{2} \)
47 \( 1 - 1.13e6iT - 5.06e11T^{2} \)
53 \( 1 - 1.19e6T + 1.17e12T^{2} \)
59 \( 1 + 7.28e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.16e6T + 3.14e12T^{2} \)
67 \( 1 + 1.54e5iT - 6.06e12T^{2} \)
71 \( 1 + 4.30e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.31e6iT - 1.10e13T^{2} \)
79 \( 1 + 4.58e6T + 1.92e13T^{2} \)
83 \( 1 + 6.49e6iT - 2.71e13T^{2} \)
89 \( 1 - 7.06e5iT - 4.42e13T^{2} \)
97 \( 1 + 4.00e6iT - 8.07e13T^{2} \)
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

−11.41361695104330547158582871781, −11.11079062210219145493641308937, −10.56965522229370394442415579860, −9.007768240840971252009612171838, −6.95801546101226965299917971661, −6.05885646347346869096616562847, −4.69363072857966159471190872234, −3.19192044200168242856216588882, −1.09303288989877264346842174384, −0.40377112185552481927292341289, 1.64893523820735574488534501975, 4.60426861933109461437473617523, 5.40449822669885765866137101904, 6.65928060847010494600865442152, 7.03168961766336900885197782498, 8.800092087542845817328383005054, 10.33812417936105969542707023256, 11.33263931525413980759386895366, 12.16150164603508903219025018750, 13.23293293868858705638608264088

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