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Properties

Label 2-69-23.6-c1-0-1
Degree $2$
Conductor $69$
Sign $0.104 - 0.994i$
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  + (−0.662 + 1.45i)2-s + (0.959 + 0.281i)3-s + (−0.354 − 0.409i)4-s + (−0.438 + 0.281i)5-s + (−1.04 + 1.20i)6-s + (0.188 − 1.31i)7-s + (−2.23 + 0.654i)8-s + (0.841 + 0.540i)9-s + (−0.118 − 0.822i)10-s + (0.0950 + 0.208i)11-s + (−0.225 − 0.492i)12-s + (−0.382 − 2.65i)13-s + (1.77 + 1.14i)14-s + (−0.499 + 0.146i)15-s + (0.681 − 4.74i)16-s + (3.83 − 4.42i)17-s + ⋯
L(s)  = 1  + (−0.468 + 1.02i)2-s + (0.553 + 0.162i)3-s + (−0.177 − 0.204i)4-s + (−0.196 + 0.125i)5-s + (−0.426 + 0.491i)6-s + (0.0712 − 0.495i)7-s + (−0.788 + 0.231i)8-s + (0.280 + 0.180i)9-s + (−0.0373 − 0.260i)10-s + (0.0286 + 0.0627i)11-s + (−0.0649 − 0.142i)12-s + (−0.105 − 0.737i)13-s + (0.474 + 0.305i)14-s + (−0.129 + 0.0379i)15-s + (0.170 − 1.18i)16-s + (0.929 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.104 - 0.994i$
Analytic conductor: \(0.550967\)
Root analytic conductor: \(0.742272\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (52, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1/2),\ 0.104 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.631055 + 0.568320i\)
\(L(\frac12)\) \(\approx\) \(0.631055 + 0.568320i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.959 - 0.281i)T \)
23 \( 1 + (0.745 + 4.73i)T \)
good2 \( 1 + (0.662 - 1.45i)T + (-1.30 - 1.51i)T^{2} \)
5 \( 1 + (0.438 - 0.281i)T + (2.07 - 4.54i)T^{2} \)
7 \( 1 + (-0.188 + 1.31i)T + (-6.71 - 1.97i)T^{2} \)
11 \( 1 + (-0.0950 - 0.208i)T + (-7.20 + 8.31i)T^{2} \)
13 \( 1 + (0.382 + 2.65i)T + (-12.4 + 3.66i)T^{2} \)
17 \( 1 + (-3.83 + 4.42i)T + (-2.41 - 16.8i)T^{2} \)
19 \( 1 + (-1.30 - 1.50i)T + (-2.70 + 18.8i)T^{2} \)
29 \( 1 + (4.19 - 4.83i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (9.52 - 2.79i)T + (26.0 - 16.7i)T^{2} \)
37 \( 1 + (-7.40 - 4.75i)T + (15.3 + 33.6i)T^{2} \)
41 \( 1 + (-0.614 + 0.395i)T + (17.0 - 37.2i)T^{2} \)
43 \( 1 + (9.98 + 2.93i)T + (36.1 + 23.2i)T^{2} \)
47 \( 1 + 2.04T + 47T^{2} \)
53 \( 1 + (-0.955 + 6.64i)T + (-50.8 - 14.9i)T^{2} \)
59 \( 1 + (-0.928 - 6.45i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-2.99 + 0.878i)T + (51.3 - 32.9i)T^{2} \)
67 \( 1 + (4.63 - 10.1i)T + (-43.8 - 50.6i)T^{2} \)
71 \( 1 + (2.37 - 5.19i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (1.53 + 1.76i)T + (-10.3 + 72.2i)T^{2} \)
79 \( 1 + (1.38 + 9.64i)T + (-75.7 + 22.2i)T^{2} \)
83 \( 1 + (0.303 + 0.194i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-14.2 - 4.17i)T + (74.8 + 48.1i)T^{2} \)
97 \( 1 + (-4.02 + 2.58i)T + (40.2 - 88.2i)T^{2} \)
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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

−14.98128338535346255438114332810, −14.42889420384629081092665598150, −13.02953701738324537990697761884, −11.67863516610654393647019539926, −10.18492646617560665619407386951, −9.014074049172183334838385567946, −7.80654276939839532533405431171, −7.11234274359037003920351567863, −5.40046090275397100460822961372, −3.29938586733823977250338832436, 1.98356146236179602396328766504, 3.69176065791913204470937826730, 5.96729268497104026043874092327, 7.76902983712440252194838208888, 9.057002484274731951050268516142, 9.867629006915898589822998131981, 11.24259925055538131213121165961, 12.08824528588528792003448105077, 13.12516824486544597310543865051, 14.54973077526827837602998908782

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