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Properties

Label 2-65-13.12-c3-0-13
Degree $2$
Conductor $65$
Sign $-0.523 - 0.851i$
Analytic cond. $3.83512$
Root an. cond. $1.95834$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.37i·2-s − 0.936·3-s − 20.9·4-s − 5i·5-s + 5.03i·6-s + 0.201i·7-s + 69.4i·8-s − 26.1·9-s − 26.8·10-s − 10.4i·11-s + 19.5·12-s + (39.9 − 24.5i)13-s + 1.08·14-s + 4.68i·15-s + 205.·16-s − 54.7·17-s + ⋯
L(s)  = 1  − 1.90i·2-s − 0.180·3-s − 2.61·4-s − 0.447i·5-s + 0.342i·6-s + 0.0108i·7-s + 3.06i·8-s − 0.967·9-s − 0.850·10-s − 0.285i·11-s + 0.471·12-s + (0.851 − 0.523i)13-s + 0.0207·14-s + 0.0806i·15-s + 3.21·16-s − 0.781·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.523 - 0.851i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.523 - 0.851i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ -0.523 - 0.851i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.356289 + 0.637259i\)
\(L(\frac12)\) \(\approx\) \(0.356289 + 0.637259i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5iT \)
13 \( 1 + (-39.9 + 24.5i)T \)
good2 \( 1 + 5.37iT - 8T^{2} \)
3 \( 1 + 0.936T + 27T^{2} \)
7 \( 1 - 0.201iT - 343T^{2} \)
11 \( 1 + 10.4iT - 1.33e3T^{2} \)
17 \( 1 + 54.7T + 4.91e3T^{2} \)
19 \( 1 + 129. iT - 6.85e3T^{2} \)
23 \( 1 + 175.T + 1.21e4T^{2} \)
29 \( 1 - 107.T + 2.43e4T^{2} \)
31 \( 1 + 22.2iT - 2.97e4T^{2} \)
37 \( 1 + 205. iT - 5.06e4T^{2} \)
41 \( 1 + 285. iT - 6.89e4T^{2} \)
43 \( 1 - 321.T + 7.95e4T^{2} \)
47 \( 1 - 379. iT - 1.03e5T^{2} \)
53 \( 1 - 506.T + 1.48e5T^{2} \)
59 \( 1 + 678. iT - 2.05e5T^{2} \)
61 \( 1 + 186.T + 2.26e5T^{2} \)
67 \( 1 - 925. iT - 3.00e5T^{2} \)
71 \( 1 - 376. iT - 3.57e5T^{2} \)
73 \( 1 + 671. iT - 3.89e5T^{2} \)
79 \( 1 + 593.T + 4.93e5T^{2} \)
83 \( 1 - 425. iT - 5.71e5T^{2} \)
89 \( 1 + 141. iT - 7.04e5T^{2} \)
97 \( 1 - 551. iT - 9.12e5T^{2} \)
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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

−13.36980568509648561006856766128, −12.30582791195698887437283114136, −11.33406895722717352785563390722, −10.62531840355555640940593384205, −9.175511252775737301623630965319, −8.423345686165744525935033397332, −5.62757424998476649755666551630, −4.13292776760020699012960293290, −2.55283591036004910893597615615, −0.51478483662979872027753307819, 4.09185949015735316092587898009, 5.76510929292569719042177829559, 6.51737936964893710187640046852, 7.923087802425750358021986109425, 8.821413583868947991386131439860, 10.23778978846328918490210062424, 11.97814965250908178862450775100, 13.60247513602621217842015241275, 14.22041961483155361301480877218, 15.17576686886318654865333826371

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