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Properties

Label 2-69-3.2-c6-0-26
Degree $2$
Conductor $69$
Sign $-0.468 + 0.883i$
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  − 8.50i·2-s + (−23.8 − 12.6i)3-s − 8.25·4-s + 65.5i·5-s + (−107. + 202. i)6-s + 552.·7-s − 473. i·8-s + (409. + 603. i)9-s + 557.·10-s − 607. i·11-s + (196. + 104. i)12-s + 3.16e3·13-s − 4.69e3i·14-s + (829. − 1.56e3i)15-s − 4.55e3·16-s + 685. i·17-s + ⋯
L(s)  = 1  − 1.06i·2-s + (−0.883 − 0.468i)3-s − 0.128·4-s + 0.524i·5-s + (−0.497 + 0.938i)6-s + 1.61·7-s − 0.925i·8-s + (0.561 + 0.827i)9-s + 0.557·10-s − 0.456i·11-s + (0.113 + 0.0604i)12-s + 1.44·13-s − 1.71i·14-s + (0.245 − 0.463i)15-s − 1.11·16-s + 0.139i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.468 + 0.883i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.468 + 0.883i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.958069 - 1.59241i\)
\(L(\frac12)\) \(\approx\) \(0.958069 - 1.59241i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (23.8 + 12.6i)T \)
23 \( 1 - 2.53e3iT \)
good2 \( 1 + 8.50iT - 64T^{2} \)
5 \( 1 - 65.5iT - 1.56e4T^{2} \)
7 \( 1 - 552.T + 1.17e5T^{2} \)
11 \( 1 + 607. iT - 1.77e6T^{2} \)
13 \( 1 - 3.16e3T + 4.82e6T^{2} \)
17 \( 1 - 685. iT - 2.41e7T^{2} \)
19 \( 1 + 4.59e3T + 4.70e7T^{2} \)
29 \( 1 + 1.98e4iT - 5.94e8T^{2} \)
31 \( 1 + 2.97e4T + 8.87e8T^{2} \)
37 \( 1 - 3.09e4T + 2.56e9T^{2} \)
41 \( 1 + 1.11e5iT - 4.75e9T^{2} \)
43 \( 1 - 3.74e4T + 6.32e9T^{2} \)
47 \( 1 + 1.05e5iT - 1.07e10T^{2} \)
53 \( 1 + 4.52e4iT - 2.21e10T^{2} \)
59 \( 1 - 2.99e5iT - 4.21e10T^{2} \)
61 \( 1 - 2.01e5T + 5.15e10T^{2} \)
67 \( 1 - 1.90e5T + 9.04e10T^{2} \)
71 \( 1 - 3.61e5iT - 1.28e11T^{2} \)
73 \( 1 + 7.07e4T + 1.51e11T^{2} \)
79 \( 1 + 2.07e5T + 2.43e11T^{2} \)
83 \( 1 - 5.11e5iT - 3.26e11T^{2} \)
89 \( 1 + 1.13e6iT - 4.96e11T^{2} \)
97 \( 1 + 1.47e6T + 8.32e11T^{2} \)
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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

−12.86157660646493845049480449284, −11.62202769057074174235655088508, −11.08594007448880621926718290894, −10.50410407605644050299615709657, −8.490036244405928575551348564769, −7.09121394455226632008955968290, −5.74766224770595361254798516579, −4.05462927259541033254128265757, −2.08348265702821215926429107428, −0.967201098135249763830932199913, 1.40735829799706213010311754172, 4.47053152688955758086010640608, 5.35011546316677412402496640363, 6.53454034725004160627344287893, 7.935779549864580637390504269587, 8.940341841753673161518217446293, 10.86925479110470904767489726367, 11.35657849095719077040711219445, 12.74475329313338470859328440607, 14.35880104882350810791131400781

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