Properties

Label 2-69-23.8-c1-0-0
Degree $2$
Conductor $69$
Sign $0.319 - 0.947i$
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  + (−0.164 + 1.14i)2-s + (0.415 + 0.909i)3-s + (0.636 + 0.186i)4-s + (−1.13 − 1.30i)5-s + (−1.10 + 0.325i)6-s + (−0.589 − 0.379i)7-s + (−1.27 + 2.80i)8-s + (−0.654 + 0.755i)9-s + (1.68 − 1.08i)10-s + (−0.485 − 3.37i)11-s + (0.0944 + 0.656i)12-s + (3.32 − 2.13i)13-s + (0.530 − 0.612i)14-s + (0.718 − 1.57i)15-s + (−1.87 − 1.20i)16-s + (0.920 − 0.270i)17-s + ⋯
L(s)  = 1  + (−0.116 + 0.809i)2-s + (0.239 + 0.525i)3-s + (0.318 + 0.0934i)4-s + (−0.506 − 0.584i)5-s + (−0.452 + 0.132i)6-s + (−0.222 − 0.143i)7-s + (−0.452 + 0.990i)8-s + (−0.218 + 0.251i)9-s + (0.532 − 0.341i)10-s + (−0.146 − 1.01i)11-s + (0.0272 + 0.189i)12-s + (0.921 − 0.592i)13-s + (0.141 − 0.163i)14-s + (0.185 − 0.406i)15-s + (−0.469 − 0.301i)16-s + (0.223 − 0.0655i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760629 + 0.546336i\)
\(L(\frac12)\) \(\approx\) \(0.760629 + 0.546336i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.415 - 0.909i)T \)
23 \( 1 + (-2.27 - 4.22i)T \)
good2 \( 1 + (0.164 - 1.14i)T + (-1.91 - 0.563i)T^{2} \)
5 \( 1 + (1.13 + 1.30i)T + (-0.711 + 4.94i)T^{2} \)
7 \( 1 + (0.589 + 0.379i)T + (2.90 + 6.36i)T^{2} \)
11 \( 1 + (0.485 + 3.37i)T + (-10.5 + 3.09i)T^{2} \)
13 \( 1 + (-3.32 + 2.13i)T + (5.40 - 11.8i)T^{2} \)
17 \( 1 + (-0.920 + 0.270i)T + (14.3 - 9.19i)T^{2} \)
19 \( 1 + (5.70 + 1.67i)T + (15.9 + 10.2i)T^{2} \)
29 \( 1 + (-2.25 + 0.661i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (3.58 - 7.85i)T + (-20.3 - 23.4i)T^{2} \)
37 \( 1 + (6.17 - 7.12i)T + (-5.26 - 36.6i)T^{2} \)
41 \( 1 + (4.08 + 4.71i)T + (-5.83 + 40.5i)T^{2} \)
43 \( 1 + (-3.32 - 7.27i)T + (-28.1 + 32.4i)T^{2} \)
47 \( 1 + 2.99T + 47T^{2} \)
53 \( 1 + (-3.96 - 2.54i)T + (22.0 + 48.2i)T^{2} \)
59 \( 1 + (-7.51 + 4.83i)T + (24.5 - 53.6i)T^{2} \)
61 \( 1 + (-3.64 + 7.97i)T + (-39.9 - 46.1i)T^{2} \)
67 \( 1 + (-0.241 + 1.67i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (1.65 - 11.5i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (-8.34 - 2.45i)T + (61.4 + 39.4i)T^{2} \)
79 \( 1 + (6.43 - 4.13i)T + (32.8 - 71.8i)T^{2} \)
83 \( 1 + (-9.33 + 10.7i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (-4.45 - 9.74i)T + (-58.2 + 67.2i)T^{2} \)
97 \( 1 + (1.30 + 1.51i)T + (-13.8 + 96.0i)T^{2} \)
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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

−15.37553505431722283168381007354, −14.14248144665601902318219879059, −12.91703211520890146822065318797, −11.52083597437003792299765905290, −10.54938790984446218984678434025, −8.706309333649692418570903581330, −8.215705594052768711476502949181, −6.61634161627239911988178186949, −5.25350465739549749246979845904, −3.38565853700966709840619747550, 2.16168548601745297890759846176, 3.78900843612807743279407843753, 6.34946477423079979463820768271, 7.31526482458421755486124778533, 8.904471464645944842557707348664, 10.30987359497568538877519234028, 11.21938801636549986282071331704, 12.26696361579018215275577058240, 13.08035278105645341599405720315, 14.65854434119570075295998473597

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