Properties

Label 2-69-23.7-c6-0-1
Degree $2$
Conductor $69$
Sign $-0.930 - 0.366i$
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  + (4.04 − 4.67i)2-s + (−13.1 + 8.42i)3-s + (3.66 + 25.5i)4-s + (−6.23 + 2.84i)5-s + (−13.7 + 95.3i)6-s + (−83.3 − 283. i)7-s + (466. + 300. i)8-s + (100. − 221. i)9-s + (−11.9 + 40.6i)10-s + (−931. + 806. i)11-s + (−263. − 303. i)12-s + (−2.35e3 − 692. i)13-s + (−1.66e3 − 759. i)14-s + (57.8 − 89.9i)15-s + (1.70e3 − 501. i)16-s + (−4.43e3 − 637. i)17-s + ⋯
L(s)  = 1  + (0.506 − 0.584i)2-s + (−0.485 + 0.312i)3-s + (0.0573 + 0.398i)4-s + (−0.0499 + 0.0227i)5-s + (−0.0634 + 0.441i)6-s + (−0.242 − 0.827i)7-s + (0.911 + 0.586i)8-s + (0.138 − 0.303i)9-s + (−0.0119 + 0.0406i)10-s + (−0.699 + 0.606i)11-s + (−0.152 − 0.175i)12-s + (−1.07 − 0.315i)13-s + (−0.605 − 0.276i)14-s + (0.0171 − 0.0266i)15-s + (0.417 − 0.122i)16-s + (−0.902 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.930 - 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.930 - 0.366i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ -0.930 - 0.366i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0356493 + 0.188000i\)
\(L(\frac12)\) \(\approx\) \(0.0356493 + 0.188000i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (13.1 - 8.42i)T \)
23 \( 1 + (1.07e4 + 5.75e3i)T \)
good2 \( 1 + (-4.04 + 4.67i)T + (-9.10 - 63.3i)T^{2} \)
5 \( 1 + (6.23 - 2.84i)T + (1.02e4 - 1.18e4i)T^{2} \)
7 \( 1 + (83.3 + 283. i)T + (-9.89e4 + 6.36e4i)T^{2} \)
11 \( 1 + (931. - 806. i)T + (2.52e5 - 1.75e6i)T^{2} \)
13 \( 1 + (2.35e3 + 692. i)T + (4.06e6 + 2.60e6i)T^{2} \)
17 \( 1 + (4.43e3 + 637. i)T + (2.31e7 + 6.80e6i)T^{2} \)
19 \( 1 + (5.38e3 - 774. i)T + (4.51e7 - 1.32e7i)T^{2} \)
29 \( 1 + (-1.59e3 + 1.11e4i)T + (-5.70e8 - 1.67e8i)T^{2} \)
31 \( 1 + (-2.31e4 - 1.48e4i)T + (3.68e8 + 8.07e8i)T^{2} \)
37 \( 1 + (431. + 196. i)T + (1.68e9 + 1.93e9i)T^{2} \)
41 \( 1 + (1.05e3 + 2.31e3i)T + (-3.11e9 + 3.58e9i)T^{2} \)
43 \( 1 + (1.66e4 + 2.58e4i)T + (-2.62e9 + 5.75e9i)T^{2} \)
47 \( 1 + 7.58e4T + 1.07e10T^{2} \)
53 \( 1 + (4.67e4 + 1.59e5i)T + (-1.86e10 + 1.19e10i)T^{2} \)
59 \( 1 + (-1.05e5 - 3.10e4i)T + (3.54e10 + 2.28e10i)T^{2} \)
61 \( 1 + (1.18e5 - 1.84e5i)T + (-2.14e10 - 4.68e10i)T^{2} \)
67 \( 1 + (-1.74e5 - 1.51e5i)T + (1.28e10 + 8.95e10i)T^{2} \)
71 \( 1 + (-3.70e5 + 4.27e5i)T + (-1.82e10 - 1.26e11i)T^{2} \)
73 \( 1 + (-8.57e4 - 5.96e5i)T + (-1.45e11 + 4.26e10i)T^{2} \)
79 \( 1 + (-1.23e5 + 4.20e5i)T + (-2.04e11 - 1.31e11i)T^{2} \)
83 \( 1 + (7.27e4 + 3.32e4i)T + (2.14e11 + 2.47e11i)T^{2} \)
89 \( 1 + (-6.77e5 - 1.05e6i)T + (-2.06e11 + 4.52e11i)T^{2} \)
97 \( 1 + (1.24e6 - 5.68e5i)T + (5.45e11 - 6.29e11i)T^{2} \)
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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

−13.65681104258988663031702827290, −12.79187154887938870340612054637, −11.88183181649460028751601943426, −10.72828141154998221351504154396, −9.927422060042964105483077965706, −8.029284273235443339208354933110, −6.87416453821809662093331559710, −4.96204786362314322727283761225, −3.96513224798992736367717896491, −2.32495273605054534977212537015, 0.06164643798268643625835531618, 2.20569293109182501651872946156, 4.58000575054997928632131493804, 5.75169481834171389836991265542, 6.62806503491183504254075033921, 8.051263882073833733114578061637, 9.670406695356372271861544266006, 10.85549270589613921071249464352, 12.09025786809695920372013404146, 13.14785219429101843555248287835

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