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Properties

Label 2-69-69.68-c7-0-6
Degree $2$
Conductor $69$
Sign $-0.998 + 0.0558i$
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  + 9.65i·2-s + (43.8 − 16.3i)3-s + 34.6·4-s − 384.·5-s + (157. + 423. i)6-s + 402. i·7-s + 1.57e3i·8-s + (1.65e3 − 1.43e3i)9-s − 3.70e3i·10-s − 4.42e3·11-s + (1.51e3 − 567. i)12-s − 8.02e3·13-s − 3.89e3·14-s + (−1.68e4 + 6.27e3i)15-s − 1.07e4·16-s + 9.36e3·17-s + ⋯
L(s)  = 1  + 0.853i·2-s + (0.936 − 0.349i)3-s + 0.271·4-s − 1.37·5-s + (0.298 + 0.799i)6-s + 0.443i·7-s + 1.08i·8-s + (0.755 − 0.655i)9-s − 1.17i·10-s − 1.00·11-s + (0.253 − 0.0947i)12-s − 1.01·13-s − 0.378·14-s + (−1.28 + 0.480i)15-s − 0.655·16-s + 0.462·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0558i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $-0.998 + 0.0558i$
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: \((2,\ 69,\ (\ :7/2),\ -0.998 + 0.0558i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.0294659 - 1.05480i\)
\(L(\frac12)\) \(\approx\) \(0.0294659 - 1.05480i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-43.8 + 16.3i)T \)
23 \( 1 + (5.57e4 + 1.73e4i)T \)
good2 \( 1 - 9.65iT - 128T^{2} \)
5 \( 1 + 384.T + 7.81e4T^{2} \)
7 \( 1 - 402. iT - 8.23e5T^{2} \)
11 \( 1 + 4.42e3T + 1.94e7T^{2} \)
13 \( 1 + 8.02e3T + 6.27e7T^{2} \)
17 \( 1 - 9.36e3T + 4.10e8T^{2} \)
19 \( 1 - 5.10e4iT - 8.93e8T^{2} \)
29 \( 1 - 2.10e5iT - 1.72e10T^{2} \)
31 \( 1 + 3.07e5T + 2.75e10T^{2} \)
37 \( 1 + 4.70e5iT - 9.49e10T^{2} \)
41 \( 1 - 2.51e5iT - 1.94e11T^{2} \)
43 \( 1 + 3.06e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.47e5iT - 5.06e11T^{2} \)
53 \( 1 - 6.38e5T + 1.17e12T^{2} \)
59 \( 1 + 3.86e5iT - 2.48e12T^{2} \)
61 \( 1 - 1.80e5iT - 3.14e12T^{2} \)
67 \( 1 - 4.90e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.73e6iT - 9.09e12T^{2} \)
73 \( 1 - 1.11e6T + 1.10e13T^{2} \)
79 \( 1 + 6.53e6iT - 1.92e13T^{2} \)
83 \( 1 - 2.52e6T + 2.71e13T^{2} \)
89 \( 1 - 5.94e6T + 4.42e13T^{2} \)
97 \( 1 - 1.17e7iT - 8.07e13T^{2} \)
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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

−14.49680070428828188656907738405, −12.62534887968191462873900866441, −11.96526937624113804232864994342, −10.41317204335360145213251881288, −8.681209715736652890398486369965, −7.73319788770102608214084241357, −7.28888411314955437358828651899, −5.49503263127163303290121376909, −3.66662383646028780682820545373, −2.19625501279138006925287566438, 0.29900925270711396037116190117, 2.35169259071307620779090464142, 3.45271575216659615704111333878, 4.58009791357683457180015041829, 7.27449767171980482103782299000, 7.88254205156580594714239615010, 9.520040345687879552216885181553, 10.54902029569585948506305358743, 11.51757584977100062995956098703, 12.58802103132540411022710672755

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