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Properties

Label 2-91-13.12-c7-0-24
Degree $2$
Conductor $91$
Sign $-0.316 - 0.948i$
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  + 20.0i·2-s − 84.8·3-s − 275.·4-s + 538. i·5-s − 1.70e3i·6-s − 343i·7-s − 2.95e3i·8-s + 5.00e3·9-s − 1.08e4·10-s − 659. i·11-s + 2.33e4·12-s + (7.51e3 − 2.50e3i)13-s + 6.88e3·14-s − 4.56e4i·15-s + 2.41e4·16-s + 3.00e4·17-s + ⋯
L(s)  = 1  + 1.77i·2-s − 1.81·3-s − 2.14·4-s + 1.92i·5-s − 3.21i·6-s − 0.377i·7-s − 2.04i·8-s + 2.28·9-s − 3.41·10-s − 0.149i·11-s + 3.89·12-s + (0.948 − 0.316i)13-s + 0.670·14-s − 3.49i·15-s + 1.47·16-s + 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(91\)    =    \(7 \cdot 13\)
Sign: $-0.316 - 0.948i$
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: \((2,\ 91,\ (\ :7/2),\ -0.316 - 0.948i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.423554 + 0.587526i\)
\(L(\frac12)\) \(\approx\) \(0.423554 + 0.587526i\)
\(L(\frac{9}{2})\) 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 \( 1 + (-7.51e3 + 2.50e3i)T \)
good2 \( 1 - 20.0iT - 128T^{2} \)
3 \( 1 + 84.8T + 2.18e3T^{2} \)
5 \( 1 - 538. iT - 7.81e4T^{2} \)
11 \( 1 + 659. iT - 1.94e7T^{2} \)
17 \( 1 - 3.00e4T + 4.10e8T^{2} \)
19 \( 1 + 1.87e4iT - 8.93e8T^{2} \)
23 \( 1 + 2.83e4T + 3.40e9T^{2} \)
29 \( 1 - 1.23e5T + 1.72e10T^{2} \)
31 \( 1 + 1.55e5iT - 2.75e10T^{2} \)
37 \( 1 + 1.65e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.34e5iT - 1.94e11T^{2} \)
43 \( 1 - 3.91e4T + 2.71e11T^{2} \)
47 \( 1 + 8.43e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.35e6T + 1.17e12T^{2} \)
59 \( 1 + 1.92e6iT - 2.48e12T^{2} \)
61 \( 1 - 3.14e5T + 3.14e12T^{2} \)
67 \( 1 - 2.15e5iT - 6.06e12T^{2} \)
71 \( 1 - 1.86e6iT - 9.09e12T^{2} \)
73 \( 1 + 4.23e6iT - 1.10e13T^{2} \)
79 \( 1 - 1.38e6T + 1.92e13T^{2} \)
83 \( 1 + 1.42e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.43e6iT - 4.42e13T^{2} \)
97 \( 1 + 3.17e6iT - 8.07e13T^{2} \)
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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

−13.36946999484879488629335336202, −11.75920463358102919427034435817, −10.72509829307647442377946987932, −9.965055187798601097484191300918, −7.78258798115666117078246245645, −6.91896808725844383684554185452, −6.21951548808393103578706677833, −5.47908975571769001633158803583, −3.85174745704632437583120678600, −0.44179402740978238292479934587, 0.962763150286462513286796428060, 1.37751543361041817364133163665, 4.02750138461561484187979409428, 4.97885830878713856303425603719, 5.82740177056658187015587484903, 8.355529200258147982261388282267, 9.568460134622949799437212710693, 10.41116947063554394373340876798, 11.64396729083626556975699980986, 12.19877999603917433589328793557

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