Properties

Label 2-69-69.68-c7-0-32
Degree $2$
Conductor $69$
Sign $0.219 + 0.975i$
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  + 16.5i·2-s + (−28.9 − 36.7i)3-s − 145.·4-s + 13.0·5-s + (607. − 479. i)6-s + 1.23e3i·7-s − 294. i·8-s + (−510. + 2.12e3i)9-s + 215. i·10-s − 3.14e3·11-s + (4.22e3 + 5.35e3i)12-s + 3.87e3·13-s − 2.03e4·14-s + (−377. − 478. i)15-s − 1.37e4·16-s + 1.48e4·17-s + ⋯
L(s)  = 1  + 1.46i·2-s + (−0.619 − 0.785i)3-s − 1.13·4-s + 0.0466·5-s + (1.14 − 0.905i)6-s + 1.35i·7-s − 0.203i·8-s + (−0.233 + 0.972i)9-s + 0.0682i·10-s − 0.712·11-s + (0.705 + 0.894i)12-s + 0.489·13-s − 1.98·14-s + (−0.0288 − 0.0366i)15-s − 0.841·16-s + 0.733·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.219 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.219 + 0.975i$
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: \((2,\ 69,\ (\ :7/2),\ 0.219 + 0.975i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.000302979 - 0.000242460i\)
\(L(\frac12)\) \(\approx\) \(0.000302979 - 0.000242460i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (28.9 + 36.7i)T \)
23 \( 1 + (5.26e4 + 2.52e4i)T \)
good2 \( 1 - 16.5iT - 128T^{2} \)
5 \( 1 - 13.0T + 7.81e4T^{2} \)
7 \( 1 - 1.23e3iT - 8.23e5T^{2} \)
11 \( 1 + 3.14e3T + 1.94e7T^{2} \)
13 \( 1 - 3.87e3T + 6.27e7T^{2} \)
17 \( 1 - 1.48e4T + 4.10e8T^{2} \)
19 \( 1 + 5.85e4iT - 8.93e8T^{2} \)
29 \( 1 - 1.59e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.15e5T + 2.75e10T^{2} \)
37 \( 1 + 2.33e5iT - 9.49e10T^{2} \)
41 \( 1 + 1.16e5iT - 1.94e11T^{2} \)
43 \( 1 + 5.34e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.83e5iT - 5.06e11T^{2} \)
53 \( 1 + 7.14e5T + 1.17e12T^{2} \)
59 \( 1 - 2.14e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.54e6iT - 3.14e12T^{2} \)
67 \( 1 + 3.93e6iT - 6.06e12T^{2} \)
71 \( 1 - 4.26e6iT - 9.09e12T^{2} \)
73 \( 1 - 2.81e5T + 1.10e13T^{2} \)
79 \( 1 - 2.99e6iT - 1.92e13T^{2} \)
83 \( 1 + 5.70e6T + 2.71e13T^{2} \)
89 \( 1 - 3.92e5T + 4.42e13T^{2} \)
97 \( 1 - 1.43e7iT - 8.07e13T^{2} \)
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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

−13.21967313178739369962969615563, −12.14074924371836590756332454407, −10.98364581242602726136698924902, −9.040943150544621101900922299865, −8.050456273568521002573313547505, −6.95174427263984549305377967351, −5.81681521756981522108505821604, −5.16382706409447132188138458129, −2.29931759357484372456772398368, −0.00015089013475445100737500106, 1.38505850816169175467876739167, 3.47198239802046293596136585655, 4.23598260325877119475866427309, 5.94561980550300547599670472294, 7.86609713174772565131980854621, 9.897948620480754303942474244178, 10.12986376322304191596812706194, 11.17523219267946154512571133398, 12.07283024462613551863896241172, 13.23970374857468767742854912130

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