Properties

Label 2-69-23.13-c5-0-17
Degree $2$
Conductor $69$
Sign $0.968 - 0.249i$
Analytic cond. $11.0664$
Root an. cond. $3.32663$
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  + (4.37 + 5.05i)2-s + (7.57 + 4.86i)3-s + (−1.80 + 12.5i)4-s + (41.9 − 91.9i)5-s + (8.56 + 59.5i)6-s + (−7.37 − 2.16i)7-s + (108. − 69.7i)8-s + (33.6 + 73.6i)9-s + (648. − 190. i)10-s + (402. − 464. i)11-s + (−74.8 + 86.4i)12-s + (−879. + 258. i)13-s + (−21.3 − 46.7i)14-s + (765. − 491. i)15-s + (1.21e3 + 357. i)16-s + (2.42 + 16.8i)17-s + ⋯
L(s)  = 1  + (0.774 + 0.893i)2-s + (0.485 + 0.312i)3-s + (−0.0565 + 0.392i)4-s + (0.751 − 1.64i)5-s + (0.0971 + 0.675i)6-s + (−0.0568 − 0.0166i)7-s + (0.599 − 0.385i)8-s + (0.138 + 0.303i)9-s + (2.05 − 0.602i)10-s + (1.00 − 1.15i)11-s + (−0.150 + 0.173i)12-s + (−1.44 + 0.423i)13-s + (−0.0290 − 0.0637i)14-s + (0.878 − 0.564i)15-s + (1.18 + 0.349i)16-s + (0.00203 + 0.0141i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.968 - 0.249i$
Analytic conductor: \(11.0664\)
Root analytic conductor: \(3.32663\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :5/2),\ 0.968 - 0.249i)\)

Particular Values

\(L(3)\) \(\approx\) \(3.44376 + 0.435861i\)
\(L(\frac12)\) \(\approx\) \(3.44376 + 0.435861i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-7.57 - 4.86i)T \)
23 \( 1 + (824. - 2.39e3i)T \)
good2 \( 1 + (-4.37 - 5.05i)T + (-4.55 + 31.6i)T^{2} \)
5 \( 1 + (-41.9 + 91.9i)T + (-2.04e3 - 2.36e3i)T^{2} \)
7 \( 1 + (7.37 + 2.16i)T + (1.41e4 + 9.08e3i)T^{2} \)
11 \( 1 + (-402. + 464. i)T + (-2.29e4 - 1.59e5i)T^{2} \)
13 \( 1 + (879. - 258. i)T + (3.12e5 - 2.00e5i)T^{2} \)
17 \( 1 + (-2.42 - 16.8i)T + (-1.36e6 + 4.00e5i)T^{2} \)
19 \( 1 + (363. - 2.52e3i)T + (-2.37e6 - 6.97e5i)T^{2} \)
29 \( 1 + (-736. - 5.12e3i)T + (-1.96e7 + 5.77e6i)T^{2} \)
31 \( 1 + (-4.27e3 + 2.74e3i)T + (1.18e7 - 2.60e7i)T^{2} \)
37 \( 1 + (-82.0 - 179. i)T + (-4.54e7 + 5.24e7i)T^{2} \)
41 \( 1 + (-3.15e3 + 6.90e3i)T + (-7.58e7 - 8.75e7i)T^{2} \)
43 \( 1 + (-1.66e4 - 1.06e4i)T + (6.10e7 + 1.33e8i)T^{2} \)
47 \( 1 + 1.55e4T + 2.29e8T^{2} \)
53 \( 1 + (-6.74e3 - 1.97e3i)T + (3.51e8 + 2.26e8i)T^{2} \)
59 \( 1 + (-1.57e4 + 4.63e3i)T + (6.01e8 - 3.86e8i)T^{2} \)
61 \( 1 + (2.53e4 - 1.63e4i)T + (3.50e8 - 7.68e8i)T^{2} \)
67 \( 1 + (2.33e4 + 2.68e4i)T + (-1.92e8 + 1.33e9i)T^{2} \)
71 \( 1 + (1.59e3 + 1.83e3i)T + (-2.56e8 + 1.78e9i)T^{2} \)
73 \( 1 + (9.38e3 - 6.52e4i)T + (-1.98e9 - 5.84e8i)T^{2} \)
79 \( 1 + (-1.49e4 + 4.39e3i)T + (2.58e9 - 1.66e9i)T^{2} \)
83 \( 1 + (1.83e4 + 4.02e4i)T + (-2.57e9 + 2.97e9i)T^{2} \)
89 \( 1 + (-4.89e4 - 3.14e4i)T + (2.31e9 + 5.07e9i)T^{2} \)
97 \( 1 + (-3.99e4 + 8.73e4i)T + (-5.62e9 - 6.48e9i)T^{2} \)
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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

−14.05042860926289756702595674170, −13.05249185581749441429982541712, −12.05051594591372566207797180175, −10.00068015856098304620781705152, −9.108582438268093499804638756717, −7.904018735804636698645239343031, −6.16220839236761837571613525080, −5.17348455385020773264019272248, −4.04336865978671702280261651665, −1.42161224067236810344290992816, 2.19166172191112104762117450623, 2.84488816872328973393259080914, 4.52328800258599949941643053490, 6.56233118175200863577288723722, 7.48019032302560672289115384318, 9.586382987101045170258376992074, 10.43562783859089739129343106311, 11.64612916485817551649132074950, 12.62359900632341691002078365359, 13.72988482051264079991415112344

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