Properties

Label 2-69-69.11-c1-0-0
Degree $2$
Conductor $69$
Sign $0.0832 - 0.996i$
Analytic cond. $0.550967$
Root an. cond. $0.742272$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.848 + 1.32i)2-s + (−1.00 + 1.41i)3-s + (−0.192 + 0.421i)4-s + (−1.50 + 0.442i)5-s + (−2.71 − 0.131i)6-s + (1.83 − 1.59i)7-s + (2.38 − 0.343i)8-s + (−0.976 − 2.83i)9-s + (−1.86 − 1.61i)10-s + (0.834 + 0.536i)11-s + (−0.400 − 0.695i)12-s + (−1.51 + 1.75i)13-s + (3.65 + 1.07i)14-s + (0.891 − 2.56i)15-s + (3.08 + 3.56i)16-s + (−1.86 − 4.07i)17-s + ⋯
L(s)  = 1  + (0.600 + 0.933i)2-s + (−0.580 + 0.814i)3-s + (−0.0962 + 0.210i)4-s + (−0.673 + 0.197i)5-s + (−1.10 − 0.0537i)6-s + (0.693 − 0.601i)7-s + (0.844 − 0.121i)8-s + (−0.325 − 0.945i)9-s + (−0.588 − 0.510i)10-s + (0.251 + 0.161i)11-s + (−0.115 − 0.200i)12-s + (−0.420 + 0.485i)13-s + (0.977 + 0.286i)14-s + (0.230 − 0.663i)15-s + (0.771 + 0.890i)16-s + (−0.451 − 0.989i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0832 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0832 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.0832 - 0.996i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.0832 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.757436 + 0.696807i\)
\(L(\frac12)\) \(\approx\) \(0.757436 + 0.696807i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.00 - 1.41i)T \)
23 \( 1 + (-4.71 + 0.883i)T \)
good2 \( 1 + (-0.848 - 1.32i)T + (-0.830 + 1.81i)T^{2} \)
5 \( 1 + (1.50 - 0.442i)T + (4.20 - 2.70i)T^{2} \)
7 \( 1 + (-1.83 + 1.59i)T + (0.996 - 6.92i)T^{2} \)
11 \( 1 + (-0.834 - 0.536i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (1.51 - 1.75i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (1.86 + 4.07i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (5.75 + 2.62i)T + (12.4 + 14.3i)T^{2} \)
29 \( 1 + (7.93 - 3.62i)T + (18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.409 + 2.84i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (0.564 - 1.92i)T + (-31.1 - 20.0i)T^{2} \)
41 \( 1 + (-2.89 - 9.84i)T + (-34.4 + 22.1i)T^{2} \)
43 \( 1 + (-10.8 - 1.56i)T + (41.2 + 12.1i)T^{2} \)
47 \( 1 + 0.223iT - 47T^{2} \)
53 \( 1 + (7.41 + 8.55i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-1.44 - 1.25i)T + (8.39 + 58.3i)T^{2} \)
61 \( 1 + (-4.00 + 0.576i)T + (58.5 - 17.1i)T^{2} \)
67 \( 1 + (0.684 + 1.06i)T + (-27.8 + 60.9i)T^{2} \)
71 \( 1 + (-0.546 - 0.850i)T + (-29.4 + 64.5i)T^{2} \)
73 \( 1 + (-3.52 + 7.71i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (5.51 + 4.77i)T + (11.2 + 78.1i)T^{2} \)
83 \( 1 + (-0.661 - 0.194i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (-0.253 + 1.76i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-2.95 - 10.0i)T + (-81.6 + 52.4i)T^{2} \)
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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

−14.94373257469054486019640830894, −14.49911637991075634868502159307, −13.05209615592090535321581628752, −11.39465625052686263202394978820, −10.86822363645448439019615100556, −9.309183981273126654944167884679, −7.56443618702132647424923883798, −6.55909452739085074353757921166, −4.96389278163695951544993945612, −4.17507961885637541407515622754, 2.08356109535688413137297885671, 4.17596689693591658371752313842, 5.65978988876295255975250787575, 7.46122320968371665344892594728, 8.473346898387887413715487269891, 10.72350076288805228195503556513, 11.36437043930853981291839498621, 12.42944086946292811531868902955, 12.80511115664477264251107561990, 14.20441888434431214063705168532

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