Properties

Label 2-69-23.11-c6-0-4
Degree $2$
Conductor $69$
Sign $-0.894 - 0.448i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−9.93 + 6.38i)2-s + (−2.21 + 15.4i)3-s + (31.3 − 68.6i)4-s + (−62.4 − 212. i)5-s + (−76.4 − 167. i)6-s + (79.5 − 68.9i)7-s + (19.2 + 133. i)8-s + (−233. − 68.4i)9-s + (1.97e3 + 1.71e3i)10-s + (−159. + 247. i)11-s + (989. + 635. i)12-s + (−1.51e3 + 1.74e3i)13-s + (−350. + 1.19e3i)14-s + (3.42e3 − 491. i)15-s + (2.11e3 + 2.44e3i)16-s + (7.92e3 − 3.61e3i)17-s + ⋯
L(s)  = 1  + (−1.24 + 0.798i)2-s + (−0.0821 + 0.571i)3-s + (0.489 − 1.07i)4-s + (−0.499 − 1.70i)5-s + (−0.354 − 0.775i)6-s + (0.232 − 0.201i)7-s + (0.0375 + 0.261i)8-s + (−0.319 − 0.0939i)9-s + (1.97 + 1.71i)10-s + (−0.119 + 0.186i)11-s + (0.572 + 0.367i)12-s + (−0.688 + 0.794i)13-s + (−0.127 + 0.434i)14-s + (1.01 − 0.145i)15-s + (0.516 + 0.596i)16-s + (1.61 − 0.736i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.448i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.894 - 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.894 - 0.448i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.894 - 0.448i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0695724 + 0.294112i\)
\(L(\frac12)\) \(\approx\) \(0.0695724 + 0.294112i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.21 - 15.4i)T \)
23 \( 1 + (7.58e3 - 9.51e3i)T \)
good2 \( 1 + (9.93 - 6.38i)T + (26.5 - 58.2i)T^{2} \)
5 \( 1 + (62.4 + 212. i)T + (-1.31e4 + 8.44e3i)T^{2} \)
7 \( 1 + (-79.5 + 68.9i)T + (1.67e4 - 1.16e5i)T^{2} \)
11 \( 1 + (159. - 247. i)T + (-7.35e5 - 1.61e6i)T^{2} \)
13 \( 1 + (1.51e3 - 1.74e3i)T + (-6.86e5 - 4.77e6i)T^{2} \)
17 \( 1 + (-7.92e3 + 3.61e3i)T + (1.58e7 - 1.82e7i)T^{2} \)
19 \( 1 + (6.42e3 + 2.93e3i)T + (3.08e7 + 3.55e7i)T^{2} \)
29 \( 1 + (2.73e3 + 5.97e3i)T + (-3.89e8 + 4.49e8i)T^{2} \)
31 \( 1 + (-4.80e3 - 3.34e4i)T + (-8.51e8 + 2.50e8i)T^{2} \)
37 \( 1 + (-373. + 1.27e3i)T + (-2.15e9 - 1.38e9i)T^{2} \)
41 \( 1 + (-5.62e4 + 1.65e4i)T + (3.99e9 - 2.56e9i)T^{2} \)
43 \( 1 + (8.51e3 + 1.22e3i)T + (6.06e9 + 1.78e9i)T^{2} \)
47 \( 1 + 4.75e4T + 1.07e10T^{2} \)
53 \( 1 + (-3.09e4 + 2.68e4i)T + (3.15e9 - 2.19e10i)T^{2} \)
59 \( 1 + (2.10e5 - 2.43e5i)T + (-6.00e9 - 4.17e10i)T^{2} \)
61 \( 1 + (-3.27e5 + 4.70e4i)T + (4.94e10 - 1.45e10i)T^{2} \)
67 \( 1 + (-1.53e5 - 2.38e5i)T + (-3.75e10 + 8.22e10i)T^{2} \)
71 \( 1 + (5.01e5 - 3.22e5i)T + (5.32e10 - 1.16e11i)T^{2} \)
73 \( 1 + (2.41e5 - 5.28e5i)T + (-9.91e10 - 1.14e11i)T^{2} \)
79 \( 1 + (-1.73e5 - 1.50e5i)T + (3.45e10 + 2.40e11i)T^{2} \)
83 \( 1 + (1.30e5 - 4.44e5i)T + (-2.75e11 - 1.76e11i)T^{2} \)
89 \( 1 + (6.04e5 + 8.69e4i)T + (4.76e11 + 1.40e11i)T^{2} \)
97 \( 1 + (1.29e5 + 4.39e5i)T + (-7.00e11 + 4.50e11i)T^{2} \)
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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

−14.34533988087037097432518251667, −12.69580407754466940619881556143, −11.69460985180638115194594304214, −10.01578767412777625032780217869, −9.228847557420540164603123541742, −8.319323210626655031731801009161, −7.34090761128485309472421107610, −5.46135063805601016955057608614, −4.24792977880760724382642277547, −1.12953006446919739220086538165, 0.21150019052963407565348208404, 2.11997975658406270170112927039, 3.24999033663213686373372587951, 6.05847319224955437526548250027, 7.59778443812813753751576823018, 8.158453353494028282759101361714, 10.02937406579477025187978129305, 10.62013791504371981392214226218, 11.61788593481354606798040257495, 12.53079638023115401217163714036

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