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Properties

Label 2-65-13.9-c3-0-3
Degree $2$
Conductor $65$
Sign $-0.572 - 0.820i$
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  + (−1.48 + 2.57i)2-s + (1.47 − 2.55i)3-s + (−0.406 − 0.704i)4-s − 5·5-s + (4.37 + 7.57i)6-s + (13.1 + 22.8i)7-s − 21.3·8-s + (9.15 + 15.8i)9-s + (7.42 − 12.8i)10-s + (−6.03 + 10.4i)11-s − 2.39·12-s + (−43.9 + 16.2i)13-s − 78.3·14-s + (−7.36 + 12.7i)15-s + (34.9 − 60.4i)16-s + (3.35 + 5.81i)17-s + ⋯
L(s)  = 1  + (−0.524 + 0.908i)2-s + (0.283 − 0.491i)3-s + (−0.0508 − 0.0880i)4-s − 0.447·5-s + (0.297 + 0.515i)6-s + (0.712 + 1.23i)7-s − 0.942·8-s + (0.339 + 0.587i)9-s + (0.234 − 0.406i)10-s + (−0.165 + 0.286i)11-s − 0.0576·12-s + (−0.938 + 0.346i)13-s − 1.49·14-s + (−0.126 + 0.219i)15-s + (0.545 − 0.945i)16-s + (0.0478 + 0.0829i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.572 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.572 - 0.820i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ -0.572 - 0.820i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.502996 + 0.964129i\)
\(L(\frac12)\) \(\approx\) \(0.502996 + 0.964129i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 5T \)
13 \( 1 + (43.9 - 16.2i)T \)
good2 \( 1 + (1.48 - 2.57i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-1.47 + 2.55i)T + (-13.5 - 23.3i)T^{2} \)
7 \( 1 + (-13.1 - 22.8i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (6.03 - 10.4i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-3.35 - 5.81i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-26.4 - 45.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-0.384 + 0.665i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-138. + 239. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 306.T + 2.97e4T^{2} \)
37 \( 1 + (145. - 251. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-106. + 184. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (105. + 183. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 525.T + 1.03e5T^{2} \)
53 \( 1 - 262.T + 1.48e5T^{2} \)
59 \( 1 + (-43.6 - 75.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (110. + 191. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-271. + 470. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-100. - 173. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 1.20e3T + 3.89e5T^{2} \)
79 \( 1 + 233.T + 4.93e5T^{2} \)
83 \( 1 - 25.0T + 5.71e5T^{2} \)
89 \( 1 + (716. - 1.24e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-484. - 839. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−15.12906598480389425205046786496, −13.94440515572873610136402455160, −12.31490627923829107545641014440, −11.78496424187555597441288527593, −9.848242478553225478728880246634, −8.366302173761896007987359388559, −7.897755043850177710455880875817, −6.62903166720156240847886542465, −5.01195037912248341281505119648, −2.42483555623194258984782980702, 0.889899256424172113089663229167, 3.16607395048334092456573905105, 4.69262989043307386222384435131, 6.94971343991330005670712105633, 8.380585124905183591917432636722, 9.736319532842850270411032550959, 10.49888206732276717011118243511, 11.46558638890755004125805061539, 12.57213366762024731455310797187, 14.15971001872038211749509476966

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