Properties

Label 2-69-23.12-c7-0-12
Degree $2$
Conductor $69$
Sign $0.997 + 0.0726i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.62 + 2.97i)2-s + (−3.84 + 26.7i)3-s + (−40.6 + 88.9i)4-s + (−310. + 91.1i)5-s + (−61.6 − 134. i)6-s + (−575. − 664. i)7-s + (−176. − 1.22e3i)8-s + (−699. − 205. i)9-s + (1.16e3 − 1.34e3i)10-s + (−1.88e3 − 1.21e3i)11-s + (−2.22e3 − 1.42e3i)12-s + (3.82e3 − 4.41e3i)13-s + (4.63e3 + 1.36e3i)14-s + (−1.24e3 − 8.64e3i)15-s + (−3.73e3 − 4.31e3i)16-s + (1.54e4 + 3.37e4i)17-s + ⋯
L(s)  = 1  + (−0.408 + 0.262i)2-s + (−0.0821 + 0.571i)3-s + (−0.317 + 0.695i)4-s + (−1.11 + 0.326i)5-s + (−0.116 − 0.255i)6-s + (−0.634 − 0.732i)7-s + (−0.121 − 0.847i)8-s + (−0.319 − 0.0939i)9-s + (0.368 − 0.424i)10-s + (−0.427 − 0.274i)11-s + (−0.371 − 0.238i)12-s + (0.482 − 0.557i)13-s + (0.451 + 0.132i)14-s + (−0.0950 − 0.661i)15-s + (−0.228 − 0.263i)16-s + (0.761 + 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0726i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.997 + 0.0726i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ 0.997 + 0.0726i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.481930 - 0.0175346i\)
\(L(\frac12)\) \(\approx\) \(0.481930 - 0.0175346i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (3.84 - 26.7i)T \)
23 \( 1 + (5.54e4 + 1.82e4i)T \)
good2 \( 1 + (4.62 - 2.97i)T + (53.1 - 116. i)T^{2} \)
5 \( 1 + (310. - 91.1i)T + (6.57e4 - 4.22e4i)T^{2} \)
7 \( 1 + (575. + 664. i)T + (-1.17e5 + 8.15e5i)T^{2} \)
11 \( 1 + (1.88e3 + 1.21e3i)T + (8.09e6 + 1.77e7i)T^{2} \)
13 \( 1 + (-3.82e3 + 4.41e3i)T + (-8.93e6 - 6.21e7i)T^{2} \)
17 \( 1 + (-1.54e4 - 3.37e4i)T + (-2.68e8 + 3.10e8i)T^{2} \)
19 \( 1 + (1.53e4 - 3.36e4i)T + (-5.85e8 - 6.75e8i)T^{2} \)
29 \( 1 + (-3.92e4 - 8.58e4i)T + (-1.12e10 + 1.30e10i)T^{2} \)
31 \( 1 + (1.98e4 + 1.38e5i)T + (-2.63e10 + 7.75e9i)T^{2} \)
37 \( 1 + (-3.73e5 - 1.09e5i)T + (7.98e10 + 5.13e10i)T^{2} \)
41 \( 1 + (-3.61e5 + 1.06e5i)T + (1.63e11 - 1.05e11i)T^{2} \)
43 \( 1 + (-7.63e4 + 5.31e5i)T + (-2.60e11 - 7.65e10i)T^{2} \)
47 \( 1 + 3.95e5T + 5.06e11T^{2} \)
53 \( 1 + (1.34e5 + 1.55e5i)T + (-1.67e11 + 1.16e12i)T^{2} \)
59 \( 1 + (-4.12e5 + 4.76e5i)T + (-3.54e11 - 2.46e12i)T^{2} \)
61 \( 1 + (3.96e5 + 2.75e6i)T + (-3.01e12 + 8.85e11i)T^{2} \)
67 \( 1 + (-1.39e6 + 8.99e5i)T + (2.51e12 - 5.51e12i)T^{2} \)
71 \( 1 + (3.08e6 - 1.98e6i)T + (3.77e12 - 8.27e12i)T^{2} \)
73 \( 1 + (-6.20e5 + 1.35e6i)T + (-7.23e12 - 8.34e12i)T^{2} \)
79 \( 1 + (-1.63e6 + 1.88e6i)T + (-2.73e12 - 1.90e13i)T^{2} \)
83 \( 1 + (-1.18e5 - 3.48e4i)T + (2.28e13 + 1.46e13i)T^{2} \)
89 \( 1 + (1.01e6 - 7.05e6i)T + (-4.24e13 - 1.24e13i)T^{2} \)
97 \( 1 + (-4.95e6 + 1.45e6i)T + (6.79e13 - 4.36e13i)T^{2} \)
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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

−13.10291426009519796116502951560, −12.22994859668756031274246630098, −10.79060290103399539599082262038, −9.951304107560670411416466342898, −8.298545398943612813766193950591, −7.76861509366880102106469906566, −6.18455734555591197026774349603, −3.97444564969608831825797218161, −3.51294394474489789024937242339, −0.30780807696571605245078969516, 0.78045943575093369106749061839, 2.60240602214190782262002314199, 4.63614939980364458405098731212, 6.04443110212567653809945157746, 7.56184980554717505115428917186, 8.771923605655131834027468692256, 9.717388186951136694388136835164, 11.28969582708881795632141584818, 11.97422646072410968029223724761, 13.18319925599616838080850209090

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