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Properties

Label 2-65-13.3-c3-0-0
Degree $2$
Conductor $65$
Sign $0.315 - 0.948i$
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  + (−1.92 − 3.33i)2-s + (4.40 + 7.62i)3-s + (−3.42 + 5.93i)4-s − 5·5-s + (16.9 − 29.3i)6-s + (−17.9 + 31.1i)7-s − 4.42·8-s + (−25.2 + 43.7i)9-s + (9.63 + 16.6i)10-s + (−7.21 − 12.4i)11-s − 60.3·12-s + (46.7 + 3.06i)13-s + 138.·14-s + (−22.0 − 38.1i)15-s + (35.9 + 62.2i)16-s + (22.4 − 38.9i)17-s + ⋯
L(s)  = 1  + (−0.681 − 1.17i)2-s + (0.847 + 1.46i)3-s + (−0.428 + 0.741i)4-s − 0.447·5-s + (1.15 − 1.99i)6-s + (−0.971 + 1.68i)7-s − 0.195·8-s + (−0.935 + 1.61i)9-s + (0.304 + 0.527i)10-s + (−0.197 − 0.342i)11-s − 1.45·12-s + (0.997 + 0.0653i)13-s + 2.64·14-s + (−0.378 − 0.656i)15-s + (0.561 + 0.972i)16-s + (0.320 − 0.555i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.315 - 0.948i$
Analytic conductor: \(3.83512\)
Root analytic conductor: \(1.95834\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :3/2),\ 0.315 - 0.948i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.754231 + 0.544168i\)
\(L(\frac12)\) \(\approx\) \(0.754231 + 0.544168i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.7 - 3.06i)T \)
good2 \( 1 + (1.92 + 3.33i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-4.40 - 7.62i)T + (-13.5 + 23.3i)T^{2} \)
7 \( 1 + (17.9 - 31.1i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (7.21 + 12.4i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-22.4 + 38.9i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-0.473 + 0.819i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-78.8 - 136. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (49.5 + 85.8i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 36.0T + 2.97e4T^{2} \)
37 \( 1 + (52.1 + 90.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (18.6 + 32.3i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (222. - 384. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 329.T + 1.03e5T^{2} \)
53 \( 1 - 493.T + 1.48e5T^{2} \)
59 \( 1 + (17.6 - 30.4i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (251. - 435. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-227. - 393. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (31.0 - 53.8i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 78.1T + 3.89e5T^{2} \)
79 \( 1 - 925.T + 4.93e5T^{2} \)
83 \( 1 + 323.T + 5.71e5T^{2} \)
89 \( 1 + (-432. - 748. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-45.1 + 78.2i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−15.03643596736352057508178116205, −13.33584648131364446224425304457, −11.94833248193322546685045939084, −11.04507503361248711081148005588, −9.818732698281489823760254408593, −9.146543404230475772576321472300, −8.457572047952223432625346274546, −5.65295457842904853601031189164, −3.56374043414121409024406686123, −2.73612307552110585012344117572, 0.72475225702553590739983772146, 3.44395636706496376295085373718, 6.46821363731700755402314244160, 7.07634751243108210740119343142, 7.930047373133694519099423573365, 8.889551930455423042100037991003, 10.45108191992115866720254152185, 12.40130961677444811353089884822, 13.29169211610237824164618812610, 14.16513858939375620098549204351

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