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Properties

Label 2-91-13.12-c7-0-5
Degree $2$
Conductor $91$
Sign $-0.780 + 0.625i$
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  + 15.1i·2-s + 24.8·3-s − 100.·4-s − 394. i·5-s + 375. i·6-s + 343i·7-s + 418. i·8-s − 1.56e3·9-s + 5.95e3·10-s − 2.20e3i·11-s − 2.49e3·12-s + (4.95e3 + 6.18e3i)13-s − 5.18e3·14-s − 9.81e3i·15-s − 1.91e4·16-s − 2.06e4·17-s + ⋯
L(s)  = 1  + 1.33i·2-s + 0.532·3-s − 0.783·4-s − 1.41i·5-s + 0.710i·6-s + 0.377i·7-s + 0.288i·8-s − 0.716·9-s + 1.88·10-s − 0.498i·11-s − 0.417·12-s + (0.625 + 0.780i)13-s − 0.504·14-s − 0.750i·15-s − 1.16·16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.780 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.780 + 0.625i$
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: \((2,\ 91,\ (\ :7/2),\ -0.780 + 0.625i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.236276 - 0.672777i\)
\(L(\frac12)\) \(\approx\) \(0.236276 - 0.672777i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-4.95e3 - 6.18e3i)T \)
good2 \( 1 - 15.1iT - 128T^{2} \)
3 \( 1 - 24.8T + 2.18e3T^{2} \)
5 \( 1 + 394. iT - 7.81e4T^{2} \)
11 \( 1 + 2.20e3iT - 1.94e7T^{2} \)
17 \( 1 + 2.06e4T + 4.10e8T^{2} \)
19 \( 1 - 5.00e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.12e5T + 3.40e9T^{2} \)
29 \( 1 + 2.56e4T + 1.72e10T^{2} \)
31 \( 1 - 8.70e4iT - 2.75e10T^{2} \)
37 \( 1 - 2.19e5iT - 9.49e10T^{2} \)
41 \( 1 + 3.38e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.33e5T + 2.71e11T^{2} \)
47 \( 1 - 3.57e5iT - 5.06e11T^{2} \)
53 \( 1 + 7.38e5T + 1.17e12T^{2} \)
59 \( 1 + 2.96e6iT - 2.48e12T^{2} \)
61 \( 1 - 6.19e5T + 3.14e12T^{2} \)
67 \( 1 + 1.84e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.84e6iT - 9.09e12T^{2} \)
73 \( 1 - 1.52e6iT - 1.10e13T^{2} \)
79 \( 1 + 1.41e6T + 1.92e13T^{2} \)
83 \( 1 - 6.09e6iT - 2.71e13T^{2} \)
89 \( 1 - 9.67e6iT - 4.42e13T^{2} \)
97 \( 1 + 1.18e7iT - 8.07e13T^{2} \)
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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

−13.83214807254374038247013770658, −12.46000121096917525437920754853, −11.36625307443446934971019967690, −9.385763017877846886162475795831, −8.454448506404182610243855227083, −8.125785413295646016634963580332, −6.28548378144650499829050853297, −5.47762126458988770937951058923, −4.07026125474127135330558888964, −1.89665848317204343829305733226, 0.19156659093801456001983201697, 2.19817278166228529923257284577, 2.94487451476188265864048575151, 4.07146406261334907629915648543, 6.29725836983124150496842736729, 7.51948442003472827775183392871, 9.031083076362901586276962588902, 10.20979156985969195637900449536, 10.96719556982237290816374191616, 11.65110285790216256309852825761

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