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Properties

Label 2-104-13.12-c9-0-15
Degree $2$
Conductor $104$
Sign $-0.741 - 0.671i$
Analytic cond. $53.5637$
Root an. cond. $7.31872$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 196.·3-s + 2.71e3i·5-s + 8.42e3i·7-s + 1.89e4·9-s + 7.63e4i·11-s + (6.91e4 − 7.63e4i)13-s + 5.33e5i·15-s + 2.76e5·17-s − 4.97e5i·19-s + 1.65e6i·21-s − 2.07e5·23-s − 5.41e6·25-s − 1.39e5·27-s − 8.17e5·29-s − 2.73e6i·31-s + ⋯
L(s)  = 1  + 1.40·3-s + 1.94i·5-s + 1.32i·7-s + 0.963·9-s + 1.57i·11-s + (0.671 − 0.741i)13-s + 2.72i·15-s + 0.804·17-s − 0.875i·19-s + 1.85i·21-s − 0.154·23-s − 2.77·25-s − 0.0504·27-s − 0.214·29-s − 0.531i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $-0.741 - 0.671i$
Analytic conductor: \(53.5637\)
Root analytic conductor: \(7.31872\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{104} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 104,\ (\ :9/2),\ -0.741 - 0.671i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.400026334\)
\(L(\frac12)\) \(\approx\) \(3.400026334\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-6.91e4 + 7.63e4i)T \)
good3 \( 1 - 196.T + 1.96e4T^{2} \)
5 \( 1 - 2.71e3iT - 1.95e6T^{2} \)
7 \( 1 - 8.42e3iT - 4.03e7T^{2} \)
11 \( 1 - 7.63e4iT - 2.35e9T^{2} \)
17 \( 1 - 2.76e5T + 1.18e11T^{2} \)
19 \( 1 + 4.97e5iT - 3.22e11T^{2} \)
23 \( 1 + 2.07e5T + 1.80e12T^{2} \)
29 \( 1 + 8.17e5T + 1.45e13T^{2} \)
31 \( 1 + 2.73e6iT - 2.64e13T^{2} \)
37 \( 1 + 2.04e7iT - 1.29e14T^{2} \)
41 \( 1 - 2.76e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.79e7T + 5.02e14T^{2} \)
47 \( 1 + 1.73e7iT - 1.11e15T^{2} \)
53 \( 1 - 9.73e7T + 3.29e15T^{2} \)
59 \( 1 + 5.24e7iT - 8.66e15T^{2} \)
61 \( 1 + 1.81e7T + 1.16e16T^{2} \)
67 \( 1 - 1.25e8iT - 2.72e16T^{2} \)
71 \( 1 + 7.75e7iT - 4.58e16T^{2} \)
73 \( 1 + 1.50e8iT - 5.88e16T^{2} \)
79 \( 1 + 8.03e6T + 1.19e17T^{2} \)
83 \( 1 - 7.41e8iT - 1.86e17T^{2} \)
89 \( 1 - 7.95e8iT - 3.50e17T^{2} \)
97 \( 1 - 2.65e8iT - 7.60e17T^{2} \)
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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

−12.48887950435204896368369699795, −11.23609577453844331084246216226, −10.05488011799804758662143057000, −9.238378960560417779961798374749, −7.924582206598377782764974361482, −7.12645996720268179736899499558, −5.79502370960719489211405188826, −3.74431256756474746710321829031, −2.66008044382713815480732220325, −2.20548315307083884945171718296, 0.73851965376934062936942564094, 1.51041507687241483380211924143, 3.49752141529223977985698084582, 4.21653728653036520951989217032, 5.75770984737159737187246548061, 7.65190799411920553567197512730, 8.479074850270574811530853989076, 9.022200076239980194022961278656, 10.23224599853831690429468392064, 11.75298814870672120057881811774

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