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Properties

Label 2-65-5.4-c5-0-22
Degree $2$
Conductor $65$
Sign $0.338 + 0.940i$
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  + 3.30i·2-s − 24.0i·3-s + 21.1·4-s + (52.5 − 18.9i)5-s + 79.3·6-s − 91.7i·7-s + 175. i·8-s − 335.·9-s + (62.5 + 173. i)10-s − 45.5·11-s − 507. i·12-s − 169i·13-s + 302.·14-s + (−455. − 1.26e3i)15-s + 96.5·16-s − 745. i·17-s + ⋯
L(s)  = 1  + 0.583i·2-s − 1.54i·3-s + 0.659·4-s + (0.940 − 0.338i)5-s + 0.900·6-s − 0.707i·7-s + 0.968i·8-s − 1.37·9-s + (0.197 + 0.549i)10-s − 0.113·11-s − 1.01i·12-s − 0.277i·13-s + 0.412·14-s + (−0.522 − 1.45i)15-s + 0.0942·16-s − 0.625i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(65\)    =    \(5 \cdot 13\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(10.4249\)
Root analytic conductor: \(3.22876\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{65} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 65,\ (\ :5/2),\ 0.338 + 0.940i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.88070 - 1.32171i\)
\(L(\frac12)\) \(\approx\) \(1.88070 - 1.32171i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-52.5 + 18.9i)T \)
13 \( 1 + 169iT \)
good2 \( 1 - 3.30iT - 32T^{2} \)
3 \( 1 + 24.0iT - 243T^{2} \)
7 \( 1 + 91.7iT - 1.68e4T^{2} \)
11 \( 1 + 45.5T + 1.61e5T^{2} \)
17 \( 1 + 745. iT - 1.41e6T^{2} \)
19 \( 1 + 306.T + 2.47e6T^{2} \)
23 \( 1 - 704. iT - 6.43e6T^{2} \)
29 \( 1 - 6.80e3T + 2.05e7T^{2} \)
31 \( 1 + 7.60e3T + 2.86e7T^{2} \)
37 \( 1 - 109. iT - 6.93e7T^{2} \)
41 \( 1 + 1.33e4T + 1.15e8T^{2} \)
43 \( 1 - 4.47e3iT - 1.47e8T^{2} \)
47 \( 1 + 4.86e3iT - 2.29e8T^{2} \)
53 \( 1 - 4.01e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.96e4T + 7.14e8T^{2} \)
61 \( 1 - 5.15e4T + 8.44e8T^{2} \)
67 \( 1 - 5.18e4iT - 1.35e9T^{2} \)
71 \( 1 - 4.22e4T + 1.80e9T^{2} \)
73 \( 1 - 4.52e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.25e4T + 3.07e9T^{2} \)
83 \( 1 + 4.69e4iT - 3.93e9T^{2} \)
89 \( 1 - 6.33e4T + 5.58e9T^{2} \)
97 \( 1 - 1.32e5iT - 8.58e9T^{2} \)
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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

−13.67083131817975682056554933462, −12.78559432997700251349324086692, −11.67795533031609564678502539496, −10.31702123562856424697504309485, −8.530202437404375390155281737468, −7.35663243721989227896902237248, −6.60637804230525373608756842810, −5.45567447632637290041855342874, −2.44756341579527529343377150973, −1.12718699783522020152161732507, 2.18026151591968824343638954773, 3.53439030119662094281475554339, 5.25798585043833352305865618742, 6.54131517139774429621699003661, 8.784416684000949074671030315493, 9.911131026899539162507751133285, 10.51410751728417083166941049063, 11.51788205285818736904899502547, 12.82533289009337421349613624747, 14.41282767779183503389903759603

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