Properties

Label 2-69-23.3-c1-0-0
Degree $2$
Conductor $69$
Sign $-0.552 - 0.833i$
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  + (0.357 + 2.48i)2-s + (−0.415 + 0.909i)3-s + (−4.14 + 1.21i)4-s + (2.65 − 3.06i)5-s + (−2.41 − 0.708i)6-s + (−1.15 + 0.744i)7-s + (−2.41 − 5.29i)8-s + (−0.654 − 0.755i)9-s + (8.57 + 5.51i)10-s + (−0.00388 + 0.0270i)11-s + (0.614 − 4.27i)12-s + (−0.527 − 0.339i)13-s + (−2.26 − 2.61i)14-s + (1.68 + 3.68i)15-s + (5.04 − 3.24i)16-s + (4.13 + 1.21i)17-s + ⋯
L(s)  = 1  + (0.252 + 1.75i)2-s + (−0.239 + 0.525i)3-s + (−2.07 + 0.608i)4-s + (1.18 − 1.37i)5-s + (−0.984 − 0.289i)6-s + (−0.437 + 0.281i)7-s + (−0.855 − 1.87i)8-s + (−0.218 − 0.251i)9-s + (2.71 + 1.74i)10-s + (−0.00117 + 0.00814i)11-s + (0.177 − 1.23i)12-s + (−0.146 − 0.0940i)13-s + (−0.605 − 0.699i)14-s + (0.434 + 0.952i)15-s + (1.26 − 0.811i)16-s + (1.00 + 0.294i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.552 - 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.552 - 0.833i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ -0.552 - 0.833i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.455386 + 0.847738i\)
\(L(\frac12)\) \(\approx\) \(0.455386 + 0.847738i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.11 + 4.66i)T \)
good2 \( 1 + (-0.357 - 2.48i)T + (-1.91 + 0.563i)T^{2} \)
5 \( 1 + (-2.65 + 3.06i)T + (-0.711 - 4.94i)T^{2} \)
7 \( 1 + (1.15 - 0.744i)T + (2.90 - 6.36i)T^{2} \)
11 \( 1 + (0.00388 - 0.0270i)T + (-10.5 - 3.09i)T^{2} \)
13 \( 1 + (0.527 + 0.339i)T + (5.40 + 11.8i)T^{2} \)
17 \( 1 + (-4.13 - 1.21i)T + (14.3 + 9.19i)T^{2} \)
19 \( 1 + (4.79 - 1.40i)T + (15.9 - 10.2i)T^{2} \)
29 \( 1 + (3.54 + 1.04i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (0.200 + 0.438i)T + (-20.3 + 23.4i)T^{2} \)
37 \( 1 + (-1.44 - 1.66i)T + (-5.26 + 36.6i)T^{2} \)
41 \( 1 + (6.56 - 7.57i)T + (-5.83 - 40.5i)T^{2} \)
43 \( 1 + (1.36 - 2.98i)T + (-28.1 - 32.4i)T^{2} \)
47 \( 1 - 8.98T + 47T^{2} \)
53 \( 1 + (-1.54 + 0.993i)T + (22.0 - 48.2i)T^{2} \)
59 \( 1 + (3.51 + 2.25i)T + (24.5 + 53.6i)T^{2} \)
61 \( 1 + (-1.92 - 4.20i)T + (-39.9 + 46.1i)T^{2} \)
67 \( 1 + (0.630 + 4.38i)T + (-64.2 + 18.8i)T^{2} \)
71 \( 1 + (-1.02 - 7.15i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (5.72 - 1.68i)T + (61.4 - 39.4i)T^{2} \)
79 \( 1 + (-1.26 - 0.815i)T + (32.8 + 71.8i)T^{2} \)
83 \( 1 + (-5.61 - 6.47i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (-5.53 + 12.1i)T + (-58.2 - 67.2i)T^{2} \)
97 \( 1 + (1.41 - 1.63i)T + (-13.8 - 96.0i)T^{2} \)
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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

−15.23415870776629258043628980528, −14.30637438326722459774532966889, −13.16425456362938904631195335022, −12.48330338645854550340218988704, −10.05004124890542602920608607187, −9.110411132293069162227463569972, −8.219636143715080311565094971812, −6.32413086973817476837253273370, −5.55893831959125006122150709896, −4.50148315843682922811825352415, 2.06804246159223874210557656941, 3.37282705052312990836249607613, 5.58770271701022535878679679935, 7.03133056147888292007657999388, 9.309431876606769252560371652490, 10.27642311071601384791173047257, 10.92621106147402566503850479687, 12.09648412693333791534261937638, 13.26286582497314835339197542608, 13.81983037085666655730390253293

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