Properties

Label 2-69-23.13-c5-0-15
Degree $2$
Conductor $69$
Sign $0.775 - 0.631i$
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  + (5.50 + 6.35i)2-s + (−7.57 − 4.86i)3-s + (−5.49 + 38.2i)4-s + (29.1 − 63.8i)5-s + (−10.7 − 74.8i)6-s + (141. + 41.4i)7-s + (−46.7 + 30.0i)8-s + (33.6 + 73.6i)9-s + (565. − 166. i)10-s + (10.9 − 12.6i)11-s + (227. − 262. i)12-s + (303. − 89.1i)13-s + (513. + 1.12e3i)14-s + (−531. + 341. i)15-s + (737. + 216. i)16-s + (155. + 1.08e3i)17-s + ⋯
L(s)  = 1  + (0.972 + 1.12i)2-s + (−0.485 − 0.312i)3-s + (−0.171 + 1.19i)4-s + (0.521 − 1.14i)5-s + (−0.122 − 0.848i)6-s + (1.08 + 0.319i)7-s + (−0.258 + 0.166i)8-s + (0.138 + 0.303i)9-s + (1.78 − 0.525i)10-s + (0.0272 − 0.0314i)11-s + (0.456 − 0.526i)12-s + (0.498 − 0.146i)13-s + (0.700 + 1.53i)14-s + (−0.609 + 0.391i)15-s + (0.720 + 0.211i)16-s + (0.130 + 0.907i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.775 - 0.631i$
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: \((2,\ 69,\ (\ :5/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.88654 + 1.02668i\)
\(L(\frac12)\) \(\approx\) \(2.88654 + 1.02668i\)
\(L(\frac{7}{2})\) 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 \( 1 + (254. + 2.52e3i)T \)
good2 \( 1 + (-5.50 - 6.35i)T + (-4.55 + 31.6i)T^{2} \)
5 \( 1 + (-29.1 + 63.8i)T + (-2.04e3 - 2.36e3i)T^{2} \)
7 \( 1 + (-141. - 41.4i)T + (1.41e4 + 9.08e3i)T^{2} \)
11 \( 1 + (-10.9 + 12.6i)T + (-2.29e4 - 1.59e5i)T^{2} \)
13 \( 1 + (-303. + 89.1i)T + (3.12e5 - 2.00e5i)T^{2} \)
17 \( 1 + (-155. - 1.08e3i)T + (-1.36e6 + 4.00e5i)T^{2} \)
19 \( 1 + (-49.2 + 342. i)T + (-2.37e6 - 6.97e5i)T^{2} \)
29 \( 1 + (326. + 2.27e3i)T + (-1.96e7 + 5.77e6i)T^{2} \)
31 \( 1 + (3.38e3 - 2.17e3i)T + (1.18e7 - 2.60e7i)T^{2} \)
37 \( 1 + (-792. - 1.73e3i)T + (-4.54e7 + 5.24e7i)T^{2} \)
41 \( 1 + (-687. + 1.50e3i)T + (-7.58e7 - 8.75e7i)T^{2} \)
43 \( 1 + (4.09e3 + 2.63e3i)T + (6.10e7 + 1.33e8i)T^{2} \)
47 \( 1 - 2.86e4T + 2.29e8T^{2} \)
53 \( 1 + (3.75e4 + 1.10e4i)T + (3.51e8 + 2.26e8i)T^{2} \)
59 \( 1 + (4.62e4 - 1.35e4i)T + (6.01e8 - 3.86e8i)T^{2} \)
61 \( 1 + (3.75e4 - 2.41e4i)T + (3.50e8 - 7.68e8i)T^{2} \)
67 \( 1 + (-255. - 294. i)T + (-1.92e8 + 1.33e9i)T^{2} \)
71 \( 1 + (-6.40e3 - 7.39e3i)T + (-2.56e8 + 1.78e9i)T^{2} \)
73 \( 1 + (4.87e3 - 3.38e4i)T + (-1.98e9 - 5.84e8i)T^{2} \)
79 \( 1 + (9.88e3 - 2.90e3i)T + (2.58e9 - 1.66e9i)T^{2} \)
83 \( 1 + (4.06e4 + 8.90e4i)T + (-2.57e9 + 2.97e9i)T^{2} \)
89 \( 1 + (-8.38e3 - 5.39e3i)T + (2.31e9 + 5.07e9i)T^{2} \)
97 \( 1 + (6.51e4 - 1.42e5i)T + (-5.62e9 - 6.48e9i)T^{2} \)
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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

−13.89100537574259343989713873637, −12.95715768445479720490719429171, −12.18492421556838487014602551950, −10.68732464296018657834050702891, −8.780447744525156909442494828522, −7.79418831376724059702624477649, −6.22127727440446114103023224014, −5.34120453906084430542715960544, −4.41600595276643757874853905456, −1.42957310927602543733459312406, 1.63768840642307007896760420474, 3.20196444481751269270993550355, 4.60754184012712322245473940296, 5.85117466668841973616362360271, 7.49633617574282721593125418332, 9.649798949848924267049276931975, 10.94471333150375382410266509936, 11.10590110224585420907030900562, 12.33661037175941862811822939503, 13.82544826666568497832009643948

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