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Properties

Label 2-104-13.12-c9-0-21
Degree $2$
Conductor $104$
Sign $0.997 + 0.0674i$
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  + 117.·3-s − 287. i·5-s + 1.03e4i·7-s − 5.79e3·9-s − 8.29e4i·11-s + (6.94e3 − 1.02e5i)13-s − 3.38e4i·15-s + 3.05e5·17-s + 3.79e5i·19-s + 1.22e6i·21-s + 8.15e5·23-s + 1.87e6·25-s − 3.00e6·27-s + 5.94e6·29-s − 2.09e6i·31-s + ⋯
L(s)  = 1  + 0.840·3-s − 0.205i·5-s + 1.63i·7-s − 0.294·9-s − 1.70i·11-s + (0.0674 − 0.997i)13-s − 0.172i·15-s + 0.887·17-s + 0.667i·19-s + 1.36i·21-s + 0.607·23-s + 0.957·25-s − 1.08·27-s + 1.56·29-s − 0.408i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(104\)    =    \(2^{3} \cdot 13\)
Sign: $0.997 + 0.0674i$
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: \((2,\ 104,\ (\ :9/2),\ 0.997 + 0.0674i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.863401844\)
\(L(\frac12)\) \(\approx\) \(2.863401844\)
\(L(\frac{11}{2})\) 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 \( 1 + (-6.94e3 + 1.02e5i)T \)
good3 \( 1 - 117.T + 1.96e4T^{2} \)
5 \( 1 + 287. iT - 1.95e6T^{2} \)
7 \( 1 - 1.03e4iT - 4.03e7T^{2} \)
11 \( 1 + 8.29e4iT - 2.35e9T^{2} \)
17 \( 1 - 3.05e5T + 1.18e11T^{2} \)
19 \( 1 - 3.79e5iT - 3.22e11T^{2} \)
23 \( 1 - 8.15e5T + 1.80e12T^{2} \)
29 \( 1 - 5.94e6T + 1.45e13T^{2} \)
31 \( 1 + 2.09e6iT - 2.64e13T^{2} \)
37 \( 1 - 2.19e5iT - 1.29e14T^{2} \)
41 \( 1 - 3.13e7iT - 3.27e14T^{2} \)
43 \( 1 - 2.58e7T + 5.02e14T^{2} \)
47 \( 1 - 5.79e6iT - 1.11e15T^{2} \)
53 \( 1 - 1.97e7T + 3.29e15T^{2} \)
59 \( 1 + 2.88e7iT - 8.66e15T^{2} \)
61 \( 1 - 2.14e8T + 1.16e16T^{2} \)
67 \( 1 + 2.28e8iT - 2.72e16T^{2} \)
71 \( 1 - 2.43e8iT - 4.58e16T^{2} \)
73 \( 1 + 1.63e8iT - 5.88e16T^{2} \)
79 \( 1 - 1.95e8T + 1.19e17T^{2} \)
83 \( 1 - 3.87e8iT - 1.86e17T^{2} \)
89 \( 1 - 4.30e8iT - 3.50e17T^{2} \)
97 \( 1 + 1.04e9iT - 7.60e17T^{2} \)
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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

−12.06338463992914136317869074276, −10.99946258950781336851930883715, −9.540486133292355680883581933745, −8.430440817659523272346721809882, −8.230794208672648178546320679611, −6.07400033305277823730718860524, −5.34581378160342898804790290636, −3.23478830650299527262783338815, −2.68992646253065019543242713266, −0.879477011663309634220343669262, 0.973413213719743173091636318756, 2.40485333239734749742868991165, 3.76389091602235497366655896084, 4.77899983183601927294997431386, 6.89283001655784208140571561277, 7.39793673550189705229575685695, 8.780595003440413626788601649172, 9.868196256463816586747300382619, 10.72786711357965469482634766340, 12.07186968087112803372478331098

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