Properties

Label 2-69-69.68-c7-0-40
Degree $2$
Conductor $69$
Sign $0.155 + 0.987i$
Analytic cond. $21.5545$
Root an. cond. $4.64268$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8.54i·2-s + (−46.3 − 5.92i)3-s + 54.9·4-s + 270.·5-s + (50.6 − 396. i)6-s − 1.35e3i·7-s + 1.56e3i·8-s + (2.11e3 + 550. i)9-s + 2.31e3i·10-s − 6.39e3·11-s + (−2.54e3 − 325. i)12-s − 1.16e4·13-s + 1.16e4·14-s + (−1.25e4 − 1.60e3i)15-s − 6.33e3·16-s + 1.95e4·17-s + ⋯
L(s)  = 1  + 0.755i·2-s + (−0.991 − 0.126i)3-s + 0.429·4-s + 0.969·5-s + (0.0958 − 0.749i)6-s − 1.49i·7-s + 1.07i·8-s + (0.967 + 0.251i)9-s + 0.732i·10-s − 1.44·11-s + (−0.425 − 0.0544i)12-s − 1.47·13-s + 1.13·14-s + (−0.961 − 0.122i)15-s − 0.386·16-s + 0.967·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.155 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.155 + 0.987i$
Analytic conductor: \(21.5545\)
Root analytic conductor: \(4.64268\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (68, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :7/2),\ 0.155 + 0.987i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.755399 - 0.645789i\)
\(L(\frac12)\) \(\approx\) \(0.755399 - 0.645789i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (46.3 + 5.92i)T \)
23 \( 1 + (1.63e4 + 5.60e4i)T \)
good2 \( 1 - 8.54iT - 128T^{2} \)
5 \( 1 - 270.T + 7.81e4T^{2} \)
7 \( 1 + 1.35e3iT - 8.23e5T^{2} \)
11 \( 1 + 6.39e3T + 1.94e7T^{2} \)
13 \( 1 + 1.16e4T + 6.27e7T^{2} \)
17 \( 1 - 1.95e4T + 4.10e8T^{2} \)
19 \( 1 + 1.87e4iT - 8.93e8T^{2} \)
29 \( 1 + 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 + 5.01e4T + 2.75e10T^{2} \)
37 \( 1 - 3.40e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.01e5iT - 1.94e11T^{2} \)
43 \( 1 + 9.67e5iT - 2.71e11T^{2} \)
47 \( 1 - 2.11e5iT - 5.06e11T^{2} \)
53 \( 1 - 6.75e5T + 1.17e12T^{2} \)
59 \( 1 - 5.32e4iT - 2.48e12T^{2} \)
61 \( 1 + 7.17e5iT - 3.14e12T^{2} \)
67 \( 1 - 2.63e6iT - 6.06e12T^{2} \)
71 \( 1 + 3.21e6iT - 9.09e12T^{2} \)
73 \( 1 + 3.57e6T + 1.10e13T^{2} \)
79 \( 1 + 6.03e6iT - 1.92e13T^{2} \)
83 \( 1 + 7.42e6T + 2.71e13T^{2} \)
89 \( 1 - 7.00e6T + 4.42e13T^{2} \)
97 \( 1 - 5.51e5iT - 8.07e13T^{2} \)
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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

−13.19373945717555677033845558025, −11.91963454477578671315244317126, −10.44457392686746225287213447109, −10.15474550109874404538576099553, −7.70272166293069151473438330324, −7.06104397220735646080957295974, −5.79184695783727772528218009358, −4.83061905752262432789332829709, −2.23525755423752767304806464764, −0.35767059483408322058470103049, 1.70961285705919411316670758393, 2.80310092053258396436779122451, 5.25287772247101892116640520933, 5.89098808503442214230140514158, 7.49831358376055546459776213181, 9.635928269009725124174007433622, 10.14203749332848640852707073842, 11.39351054171249708643300100873, 12.38089793935324057729221622196, 12.86169729851289218060058802647

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