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Properties

Label 2-65-5.4-c3-0-11
Degree $2$
Conductor $65$
Sign $0.983 + 0.178i$
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  + 3i·2-s − 4i·3-s − 4-s + (2 − 11i)5-s + 12·6-s − 28i·7-s + 21i·8-s + 11·9-s + (33 + 6i)10-s + 2·11-s + 4i·12-s + 13i·13-s + 84·14-s + (−44 − 8i)15-s − 71·16-s + 44i·17-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.769i·3-s − 0.125·4-s + (0.178 − 0.983i)5-s + 0.816·6-s − 1.51i·7-s + 0.928i·8-s + 0.407·9-s + (1.04 + 0.189i)10-s + 0.0548·11-s + 0.0962i·12-s + 0.277i·13-s + 1.60·14-s + (−0.757 − 0.137i)15-s − 1.10·16-s + 0.627i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.983 + 0.178i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ 0.983 + 0.178i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.61502 - 0.145626i\)
\(L(\frac12)\) \(\approx\) \(1.61502 - 0.145626i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2 + 11i)T \)
13 \( 1 - 13iT \)
good2 \( 1 - 3iT - 8T^{2} \)
3 \( 1 + 4iT - 27T^{2} \)
7 \( 1 + 28iT - 343T^{2} \)
11 \( 1 - 2T + 1.33e3T^{2} \)
17 \( 1 - 44iT - 4.91e3T^{2} \)
19 \( 1 - 94T + 6.85e3T^{2} \)
23 \( 1 - 18iT - 1.21e4T^{2} \)
29 \( 1 + 118T + 2.43e4T^{2} \)
31 \( 1 + 100T + 2.97e4T^{2} \)
37 \( 1 - 126iT - 5.06e4T^{2} \)
41 \( 1 - 474T + 6.89e4T^{2} \)
43 \( 1 - 200iT - 7.95e4T^{2} \)
47 \( 1 - 448iT - 1.03e5T^{2} \)
53 \( 1 - 754iT - 1.48e5T^{2} \)
59 \( 1 - 446T + 2.05e5T^{2} \)
61 \( 1 + 638T + 2.26e5T^{2} \)
67 \( 1 + 868iT - 3.00e5T^{2} \)
71 \( 1 - 536T + 3.57e5T^{2} \)
73 \( 1 - 58iT - 3.89e5T^{2} \)
79 \( 1 + 232T + 4.93e5T^{2} \)
83 \( 1 - 108iT - 5.71e5T^{2} \)
89 \( 1 + 1.03e3T + 7.04e5T^{2} \)
97 \( 1 + 774iT - 9.12e5T^{2} \)
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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

−14.19554265892579267091194191455, −13.48318087477544236313642612609, −12.46279077856570265419521177725, −11.02821297815105599408245436464, −9.476083471437245944366144140407, −7.87148909525416711035126799661, −7.29989652488212901867624016780, −6.01552443344926889938851625911, −4.41769976519254993054488487677, −1.31667831868911374693673907045, 2.32078782617385867925296208613, 3.55221695598140493293518624614, 5.53158983050444238423282473105, 7.15551933336132191915855886003, 9.235812223231430245681938548561, 9.970767588384887599836808890214, 11.07145675670139646443498833400, 11.87225947776325329620607463824, 13.01433643672460435022811115141, 14.57692156642353050120083977909

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