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Properties

Label 2-69-69.68-c7-0-25
Degree $2$
Conductor $69$
Sign $0.428 - 0.903i$
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  + 11.0i·2-s + (16.6 − 43.7i)3-s + 5.64·4-s + 185.·5-s + (483. + 184. i)6-s + 711. i·7-s + 1.47e3i·8-s + (−1.63e3 − 1.45e3i)9-s + 2.05e3i·10-s − 1.19e3·11-s + (93.8 − 246. i)12-s + 4.72e3·13-s − 7.86e3·14-s + (3.09e3 − 8.11e3i)15-s − 1.56e4·16-s + 4.00e4·17-s + ⋯
L(s)  = 1  + 0.977i·2-s + (0.355 − 0.934i)3-s + 0.0440·4-s + 0.664·5-s + (0.913 + 0.347i)6-s + 0.783i·7-s + 1.02i·8-s + (−0.746 − 0.665i)9-s + 0.649i·10-s − 0.271·11-s + (0.0156 − 0.0411i)12-s + 0.596·13-s − 0.766·14-s + (0.236 − 0.621i)15-s − 0.953·16-s + 1.97·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.428 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.428 - 0.903i$
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: \((2,\ 69,\ (\ :7/2),\ 0.428 - 0.903i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.35307 + 1.48862i\)
\(L(\frac12)\) \(\approx\) \(2.35307 + 1.48862i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-16.6 + 43.7i)T \)
23 \( 1 + (-5.81e4 - 4.59e3i)T \)
good2 \( 1 - 11.0iT - 128T^{2} \)
5 \( 1 - 185.T + 7.81e4T^{2} \)
7 \( 1 - 711. iT - 8.23e5T^{2} \)
11 \( 1 + 1.19e3T + 1.94e7T^{2} \)
13 \( 1 - 4.72e3T + 6.27e7T^{2} \)
17 \( 1 - 4.00e4T + 4.10e8T^{2} \)
19 \( 1 - 3.77e4iT - 8.93e8T^{2} \)
29 \( 1 + 1.82e5iT - 1.72e10T^{2} \)
31 \( 1 - 1.27e5T + 2.75e10T^{2} \)
37 \( 1 - 1.56e5iT - 9.49e10T^{2} \)
41 \( 1 + 2.35e5iT - 1.94e11T^{2} \)
43 \( 1 - 8.00e5iT - 2.71e11T^{2} \)
47 \( 1 - 9.23e5iT - 5.06e11T^{2} \)
53 \( 1 - 9.46e5T + 1.17e12T^{2} \)
59 \( 1 + 1.27e6iT - 2.48e12T^{2} \)
61 \( 1 + 1.65e6iT - 3.14e12T^{2} \)
67 \( 1 + 9.50e5iT - 6.06e12T^{2} \)
71 \( 1 - 1.41e5iT - 9.09e12T^{2} \)
73 \( 1 + 2.82e6T + 1.10e13T^{2} \)
79 \( 1 + 4.07e6iT - 1.92e13T^{2} \)
83 \( 1 + 8.08e6T + 2.71e13T^{2} \)
89 \( 1 + 6.97e6T + 4.42e13T^{2} \)
97 \( 1 - 6.73e6iT - 8.07e13T^{2} \)
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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

−13.77959165470569260795527488101, −12.51947982099533800119316333376, −11.54338672177426688697709373856, −9.782418715909495607888065701279, −8.366122553523980128793703284951, −7.63743354841337489830155121542, −6.14524185589218132291892743640, −5.65703791420906595206100758861, −2.87628298648502251703891894365, −1.49966006072399559796532709865, 1.05665188186251370086755748967, 2.73972065863716903431084547265, 3.80830983402653079993288016269, 5.40993762349595254192469407776, 7.19846175082237692077748552688, 8.915376252633340352127682875603, 10.10462623928048100405392191304, 10.55633669163710121998713601898, 11.69460632792967561524524247316, 13.14127144250120236853477915715

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