Properties

Label 2-69-23.22-c4-0-14
Degree $2$
Conductor $69$
Sign $0.163 + 0.986i$
Analytic cond. $7.13252$
Root an. cond. $2.67067$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.07·2-s + 5.19·3-s − 6.55·4-s − 40.4i·5-s + 15.9·6-s − 40.5i·7-s − 69.3·8-s + 27·9-s − 124. i·10-s − 11.5i·11-s − 34.0·12-s + 260.·13-s − 124. i·14-s − 210. i·15-s − 108.·16-s + 271. i·17-s + ⋯
L(s)  = 1  + 0.768·2-s + 0.577·3-s − 0.409·4-s − 1.61i·5-s + 0.443·6-s − 0.827i·7-s − 1.08·8-s + 0.333·9-s − 1.24i·10-s − 0.0954i·11-s − 0.236·12-s + 1.53·13-s − 0.635i·14-s − 0.935i·15-s − 0.422·16-s + 0.938i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.163 + 0.986i$
Analytic conductor: \(7.13252\)
Root analytic conductor: \(2.67067\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :2),\ 0.163 + 0.986i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.77532 - 1.50585i\)
\(L(\frac12)\) \(\approx\) \(1.77532 - 1.50585i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 5.19T \)
23 \( 1 + (86.3 + 521. i)T \)
good2 \( 1 - 3.07T + 16T^{2} \)
5 \( 1 + 40.4iT - 625T^{2} \)
7 \( 1 + 40.5iT - 2.40e3T^{2} \)
11 \( 1 + 11.5iT - 1.46e4T^{2} \)
13 \( 1 - 260.T + 2.85e4T^{2} \)
17 \( 1 - 271. iT - 8.35e4T^{2} \)
19 \( 1 - 100. iT - 1.30e5T^{2} \)
29 \( 1 - 326.T + 7.07e5T^{2} \)
31 \( 1 - 691.T + 9.23e5T^{2} \)
37 \( 1 - 484. iT - 1.87e6T^{2} \)
41 \( 1 + 1.89e3T + 2.82e6T^{2} \)
43 \( 1 - 2.55e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.13e3T + 4.87e6T^{2} \)
53 \( 1 + 3.57e3iT - 7.89e6T^{2} \)
59 \( 1 - 2.94e3T + 1.21e7T^{2} \)
61 \( 1 - 6.52e3iT - 1.38e7T^{2} \)
67 \( 1 - 120. iT - 2.01e7T^{2} \)
71 \( 1 - 3.64e3T + 2.54e7T^{2} \)
73 \( 1 + 5.41e3T + 2.83e7T^{2} \)
79 \( 1 + 1.01e4iT - 3.89e7T^{2} \)
83 \( 1 + 9.95e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.87e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.58e3iT - 8.85e7T^{2} \)
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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

−13.38978204939749421092372290696, −13.14780013929484518734486063343, −11.98789991559493980890311997583, −10.22456783108299822483029103732, −8.789467749102414280204550281846, −8.296473202891895135958924520140, −6.10354621136291669121873462000, −4.62202389634766299272449404271, −3.76294228462132632951801939554, −1.03726597528361878206321902813, 2.71111186841719775240795449719, 3.74200049072559460216629378782, 5.64970105303830110086852657672, 6.86371067966475054160966550359, 8.484002657892773712264685550691, 9.636928483584879175533067687337, 11.03803146014928624338330457362, 12.13344946374341095497583316726, 13.71185856835042368512840924361, 13.92556565449547355308483422829

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