Properties

Label 2-69-23.18-c1-0-2
Degree $2$
Conductor $69$
Sign $0.881 - 0.471i$
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  + (1.54 + 0.454i)2-s + (−0.654 + 0.755i)3-s + (0.506 + 0.325i)4-s + (0.0749 − 0.521i)5-s + (−1.35 + 0.872i)6-s + (0.200 + 0.439i)7-s + (−1.47 − 1.70i)8-s + (−0.142 − 0.989i)9-s + (0.353 − 0.772i)10-s + (−3.21 + 0.943i)11-s + (−0.577 + 0.169i)12-s + (0.853 − 1.86i)13-s + (0.111 + 0.772i)14-s + (0.344 + 0.398i)15-s + (−2.01 − 4.40i)16-s + (−0.919 + 0.590i)17-s + ⋯
L(s)  = 1  + (1.09 + 0.321i)2-s + (−0.378 + 0.436i)3-s + (0.253 + 0.162i)4-s + (0.0335 − 0.233i)5-s + (−0.554 + 0.356i)6-s + (0.0759 + 0.166i)7-s + (−0.522 − 0.602i)8-s + (−0.0474 − 0.329i)9-s + (0.111 − 0.244i)10-s + (−0.968 + 0.284i)11-s + (−0.166 + 0.0489i)12-s + (0.236 − 0.518i)13-s + (0.0296 + 0.206i)14-s + (0.0890 + 0.102i)15-s + (−0.502 − 1.10i)16-s + (−0.223 + 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.881 - 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20569 + 0.302335i\)
\(L(\frac12)\) \(\approx\) \(1.20569 + 0.302335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (1.81 - 4.43i)T \)
good2 \( 1 + (-1.54 - 0.454i)T + (1.68 + 1.08i)T^{2} \)
5 \( 1 + (-0.0749 + 0.521i)T + (-4.79 - 1.40i)T^{2} \)
7 \( 1 + (-0.200 - 0.439i)T + (-4.58 + 5.29i)T^{2} \)
11 \( 1 + (3.21 - 0.943i)T + (9.25 - 5.94i)T^{2} \)
13 \( 1 + (-0.853 + 1.86i)T + (-8.51 - 9.82i)T^{2} \)
17 \( 1 + (0.919 - 0.590i)T + (7.06 - 15.4i)T^{2} \)
19 \( 1 + (-6.19 - 3.98i)T + (7.89 + 17.2i)T^{2} \)
29 \( 1 + (3.03 - 1.94i)T + (12.0 - 26.3i)T^{2} \)
31 \( 1 + (3.65 + 4.21i)T + (-4.41 + 30.6i)T^{2} \)
37 \( 1 + (-0.491 - 3.42i)T + (-35.5 + 10.4i)T^{2} \)
41 \( 1 + (-1.47 + 10.2i)T + (-39.3 - 11.5i)T^{2} \)
43 \( 1 + (5.05 - 5.83i)T + (-6.11 - 42.5i)T^{2} \)
47 \( 1 - 10.7T + 47T^{2} \)
53 \( 1 + (3.06 + 6.71i)T + (-34.7 + 40.0i)T^{2} \)
59 \( 1 + (0.390 - 0.854i)T + (-38.6 - 44.5i)T^{2} \)
61 \( 1 + (3.75 + 4.33i)T + (-8.68 + 60.3i)T^{2} \)
67 \( 1 + (12.3 + 3.62i)T + (56.3 + 36.2i)T^{2} \)
71 \( 1 + (5.99 + 1.76i)T + (59.7 + 38.3i)T^{2} \)
73 \( 1 + (0.00713 + 0.00458i)T + (30.3 + 66.4i)T^{2} \)
79 \( 1 + (-4.29 + 9.40i)T + (-51.7 - 59.7i)T^{2} \)
83 \( 1 + (-0.861 - 5.99i)T + (-79.6 + 23.3i)T^{2} \)
89 \( 1 + (1.84 - 2.13i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (2.61 - 18.1i)T + (-93.0 - 27.3i)T^{2} \)
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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

−14.91976514638015796662435516045, −13.72186822864152902139386639478, −12.82892719409696373130837377377, −11.83638940743638968527939037047, −10.41450681127969300118964260460, −9.257485098595113167663307937036, −7.52309602404587436028351821403, −5.79231152361413839361787310080, −5.08046347029866946072625794704, −3.52415423486417004343210322678, 2.82825597089822564979806938450, 4.63863833419681775585097057188, 5.83953416083387693711026092355, 7.31735970429080944968569188573, 8.856903819043072289275027763916, 10.66865575599870260615199888474, 11.58057261465295193777320355147, 12.62323354203579353470263275211, 13.53490981853323164927861101674, 14.23432630183480271372951477654

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