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Properties

Label 2-65-13.10-c5-0-21
Degree $2$
Conductor $65$
Sign $-0.592 + 0.805i$
Analytic cond. $10.4249$
Root an. cond. $3.22876$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.88 − 4.55i)2-s + (−0.881 − 1.52i)3-s + (25.4 − 44.0i)4-s − 25i·5-s + (−13.8 − 8.02i)6-s + (−168. − 97.1i)7-s − 171. i·8-s + (119. − 207. i)9-s + (−113. − 197. i)10-s + (−1.47 + 0.848i)11-s − 89.6·12-s + (606. + 59.7i)13-s − 1.76e3·14-s + (−38.1 + 22.0i)15-s + (32.2 + 55.8i)16-s + (432. − 748. i)17-s + ⋯
L(s)  = 1  + (1.39 − 0.804i)2-s + (−0.0565 − 0.0979i)3-s + (0.794 − 1.37i)4-s − 0.447i·5-s + (−0.157 − 0.0909i)6-s + (−1.29 − 0.749i)7-s − 0.948i·8-s + (0.493 − 0.854i)9-s + (−0.359 − 0.623i)10-s + (−0.00366 + 0.00211i)11-s − 0.179·12-s + (0.995 + 0.0980i)13-s − 2.41·14-s + (−0.0438 + 0.0252i)15-s + (0.0314 + 0.0545i)16-s + (0.362 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $-0.592 + 0.805i$
Analytic conductor: \(10.4249\)
Root analytic conductor: \(3.22876\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :5/2),\ -0.592 + 0.805i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.44315 - 2.85474i\)
\(L(\frac12)\) \(\approx\) \(1.44315 - 2.85474i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 25iT \)
13 \( 1 + (-606. - 59.7i)T \)
good2 \( 1 + (-7.88 + 4.55i)T + (16 - 27.7i)T^{2} \)
3 \( 1 + (0.881 + 1.52i)T + (-121.5 + 210. i)T^{2} \)
7 \( 1 + (168. + 97.1i)T + (8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (1.47 - 0.848i)T + (8.05e4 - 1.39e5i)T^{2} \)
17 \( 1 + (-432. + 748. i)T + (-7.09e5 - 1.22e6i)T^{2} \)
19 \( 1 + (541. + 312. i)T + (1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.42e3 - 2.47e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (557. + 965. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 - 8.73e3iT - 2.86e7T^{2} \)
37 \( 1 + (-8.07e3 + 4.66e3i)T + (3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (-1.56e4 + 9.01e3i)T + (5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (546. - 947. i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + 1.49e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.82e4T + 4.18e8T^{2} \)
59 \( 1 + (-4.15e4 - 2.39e4i)T + (3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (1.34e4 - 2.33e4i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (7.06e3 - 4.07e3i)T + (6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (4.01e3 + 2.32e3i)T + (9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + 5.93e3iT - 2.07e9T^{2} \)
79 \( 1 + 7.94e4T + 3.07e9T^{2} \)
83 \( 1 + 4.80e4iT - 3.93e9T^{2} \)
89 \( 1 + (6.73e4 - 3.88e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (1.26e5 + 7.32e4i)T + (4.29e9 + 7.43e9i)T^{2} \)
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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

−13.18508774805419323318380998825, −12.71596347276398583055672006401, −11.59341833956440402011303115954, −10.37382600276267036303591998025, −9.194631818163093083171214293213, −6.97216238219383445506935596686, −5.79621912958783170846006114410, −4.15305734076197486567438972601, −3.23476446727541988247420198865, −1.04118951929445600675065027993, 2.85825124085300052264180268014, 4.21342564871520698589646123759, 5.83138749303358736727396487373, 6.51864411475177845134944572973, 7.970966784389868936863672072684, 9.740822273298691673913120486724, 11.15046607228703886609315026719, 12.80117324675582215201008573300, 13.02303855902013057110415339644, 14.34661069082577803358672282387

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