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Properties

Label 2-91-13.12-c7-0-41
Degree $2$
Conductor $91$
Sign $0.668 + 0.744i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0356i·2-s + 75.1·3-s + 127.·4-s − 480. i·5-s − 2.68i·6-s + 343i·7-s − 9.13i·8-s + 3.46e3·9-s − 17.1·10-s + 3.11e3i·11-s + 9.62e3·12-s + (5.89e3 − 5.29e3i)13-s + 12.2·14-s − 3.61e4i·15-s + 1.63e4·16-s − 1.78e4·17-s + ⋯
L(s)  = 1  − 0.00315i·2-s + 1.60·3-s + 0.999·4-s − 1.72i·5-s − 0.00507i·6-s + 0.377i·7-s − 0.00630i·8-s + 1.58·9-s − 0.00542·10-s + 0.706i·11-s + 1.60·12-s + (0.744 − 0.668i)13-s + 0.00119·14-s − 2.76i·15-s + 0.999·16-s − 0.879·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $0.668 + 0.744i$
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: \((2,\ 91,\ (\ :7/2),\ 0.668 + 0.744i)\)

Particular Values

\(L(4)\) \(\approx\) \(4.08753 - 1.82324i\)
\(L(\frac12)\) \(\approx\) \(4.08753 - 1.82324i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-5.89e3 + 5.29e3i)T \)
good2 \( 1 + 0.0356iT - 128T^{2} \)
3 \( 1 - 75.1T + 2.18e3T^{2} \)
5 \( 1 + 480. iT - 7.81e4T^{2} \)
11 \( 1 - 3.11e3iT - 1.94e7T^{2} \)
17 \( 1 + 1.78e4T + 4.10e8T^{2} \)
19 \( 1 - 5.83e3iT - 8.93e8T^{2} \)
23 \( 1 - 6.66e4T + 3.40e9T^{2} \)
29 \( 1 + 1.13e5T + 1.72e10T^{2} \)
31 \( 1 - 2.86e4iT - 2.75e10T^{2} \)
37 \( 1 - 9.45e4iT - 9.49e10T^{2} \)
41 \( 1 + 3.48e5iT - 1.94e11T^{2} \)
43 \( 1 + 8.62e5T + 2.71e11T^{2} \)
47 \( 1 + 8.23e5iT - 5.06e11T^{2} \)
53 \( 1 - 8.92e5T + 1.17e12T^{2} \)
59 \( 1 - 1.70e6iT - 2.48e12T^{2} \)
61 \( 1 - 2.78e6T + 3.14e12T^{2} \)
67 \( 1 - 2.99e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.40e6iT - 9.09e12T^{2} \)
73 \( 1 - 4.01e6iT - 1.10e13T^{2} \)
79 \( 1 - 1.07e6T + 1.92e13T^{2} \)
83 \( 1 + 9.49e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.14e6iT - 4.42e13T^{2} \)
97 \( 1 - 9.67e6iT - 8.07e13T^{2} \)
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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

−12.88433136433646760113157565645, −11.67615389998008370056460441122, −10.04616195160801370176095448931, −8.854004077706752695469832008473, −8.368563121274902338539626303821, −7.14007529783029377095021169901, −5.33573362808916742603469477250, −3.79988999039285861148507623031, −2.33615948760616866752840443769, −1.30259296545938820183380483528, 1.88369337544828635523701152466, 2.92940655689198690964943501893, 3.64608975823994281173285592482, 6.44726763620317850245496653843, 7.12575943212317714871422135944, 8.188247101050224205487716252930, 9.513567602372444201708633658093, 10.82958891064012791755150671546, 11.29627888757277712283589663856, 13.27209027710898620796263558072

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