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Properties

Label 2-91-13.12-c7-0-28
Degree $2$
Conductor $91$
Sign $-0.989 - 0.144i$
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  − 14.7i·2-s − 70.7·3-s − 88.5·4-s + 406. i·5-s + 1.04e3i·6-s + 343i·7-s − 579. i·8-s + 2.82e3·9-s + 5.98e3·10-s + 1.59e3i·11-s + 6.26e3·12-s + (−1.14e3 + 7.83e3i)13-s + 5.04e3·14-s − 2.87e4i·15-s − 1.98e4·16-s + 1.09e4·17-s + ⋯
L(s)  = 1  − 1.30i·2-s − 1.51·3-s − 0.692·4-s + 1.45i·5-s + 1.96i·6-s + 0.377i·7-s − 0.400i·8-s + 1.28·9-s + 1.89·10-s + 0.362i·11-s + 1.04·12-s + (−0.144 + 0.989i)13-s + 0.491·14-s − 2.20i·15-s − 1.21·16-s + 0.539·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.989 - 0.144i$
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: \((2,\ 91,\ (\ :7/2),\ -0.989 - 0.144i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0163090 + 0.224752i\)
\(L(\frac12)\) \(\approx\) \(0.0163090 + 0.224752i\)
\(L(\frac{9}{2})\) 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 \( 1 + (1.14e3 - 7.83e3i)T \)
good2 \( 1 + 14.7iT - 128T^{2} \)
3 \( 1 + 70.7T + 2.18e3T^{2} \)
5 \( 1 - 406. iT - 7.81e4T^{2} \)
11 \( 1 - 1.59e3iT - 1.94e7T^{2} \)
17 \( 1 - 1.09e4T + 4.10e8T^{2} \)
19 \( 1 + 3.68e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.88e4T + 3.40e9T^{2} \)
29 \( 1 + 6.05e4T + 1.72e10T^{2} \)
31 \( 1 - 1.90e5iT - 2.75e10T^{2} \)
37 \( 1 - 1.46e4iT - 9.49e10T^{2} \)
41 \( 1 - 1.83e5iT - 1.94e11T^{2} \)
43 \( 1 - 4.12e4T + 2.71e11T^{2} \)
47 \( 1 + 8.09e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.70e6T + 1.17e12T^{2} \)
59 \( 1 + 1.81e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.46e5T + 3.14e12T^{2} \)
67 \( 1 + 3.35e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.68e6iT - 9.09e12T^{2} \)
73 \( 1 + 5.15e6iT - 1.10e13T^{2} \)
79 \( 1 + 7.23e6T + 1.92e13T^{2} \)
83 \( 1 - 6.28e6iT - 2.71e13T^{2} \)
89 \( 1 + 6.71e6iT - 4.42e13T^{2} \)
97 \( 1 - 4.91e6iT - 8.07e13T^{2} \)
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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

−11.75388754501335285883259627838, −11.15091377173671586998148742703, −10.46674830217765610300191619921, −9.474658922958587596022939865851, −7.06507669778426791328989357783, −6.36600865179953463952162974714, −4.75741241379023167929547787597, −3.18491346111939644225917540201, −1.82805370637780146741021776867, −0.10272700555163550520393904785, 1.07873428259705631116927368611, 4.41229514851371716469741769458, 5.55788035338694771365702341137, 5.90776815554766614253212336491, 7.48793176859220826923685019915, 8.406824539017882103462827006158, 9.926437863947298529645652310171, 11.22688022250910093513864818902, 12.27676881024472024354825402527, 13.10125123394161576757370203372

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