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Properties

Label 2-69-23.22-c6-0-9
Degree $2$
Conductor $69$
Sign $0.355 + 0.934i$
Analytic cond. $15.8737$
Root an. cond. $3.98418$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 13.6·2-s − 15.5·3-s + 123.·4-s − 37.2i·5-s + 213.·6-s − 322. i·7-s − 806.·8-s + 243·9-s + 509. i·10-s + 2.21e3i·11-s − 1.91e3·12-s − 449.·13-s + 4.41e3i·14-s + 581. i·15-s + 3.16e3·16-s + 5.35e3i·17-s + ⋯
L(s)  = 1  − 1.70·2-s − 0.577·3-s + 1.92·4-s − 0.298i·5-s + 0.986·6-s − 0.941i·7-s − 1.57·8-s + 0.333·9-s + 0.509i·10-s + 1.66i·11-s − 1.10·12-s − 0.204·13-s + 1.60i·14-s + 0.172i·15-s + 0.771·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.355 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.355 + 0.934i$
Analytic conductor: \(15.8737\)
Root analytic conductor: \(3.98418\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :3),\ 0.355 + 0.934i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.404522 - 0.278973i\)
\(L(\frac12)\) \(\approx\) \(0.404522 - 0.278973i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 15.5T \)
23 \( 1 + (4.32e3 + 1.13e4i)T \)
good2 \( 1 + 13.6T + 64T^{2} \)
5 \( 1 + 37.2iT - 1.56e4T^{2} \)
7 \( 1 + 322. iT - 1.17e5T^{2} \)
11 \( 1 - 2.21e3iT - 1.77e6T^{2} \)
13 \( 1 + 449.T + 4.82e6T^{2} \)
17 \( 1 - 5.35e3iT - 2.41e7T^{2} \)
19 \( 1 + 2.64e3iT - 4.70e7T^{2} \)
29 \( 1 - 3.29e3T + 5.94e8T^{2} \)
31 \( 1 - 2.18e4T + 8.87e8T^{2} \)
37 \( 1 + 3.69e4iT - 2.56e9T^{2} \)
41 \( 1 + 8.41e4T + 4.75e9T^{2} \)
43 \( 1 + 2.36e4iT - 6.32e9T^{2} \)
47 \( 1 - 6.58e4T + 1.07e10T^{2} \)
53 \( 1 + 1.83e5iT - 2.21e10T^{2} \)
59 \( 1 - 2.83e5T + 4.21e10T^{2} \)
61 \( 1 + 3.60e4iT - 5.15e10T^{2} \)
67 \( 1 + 1.54e5iT - 9.04e10T^{2} \)
71 \( 1 + 4.15e5T + 1.28e11T^{2} \)
73 \( 1 + 4.46e4T + 1.51e11T^{2} \)
79 \( 1 + 7.51e5iT - 2.43e11T^{2} \)
83 \( 1 + 9.88e5iT - 3.26e11T^{2} \)
89 \( 1 + 4.10e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.20e6iT - 8.32e11T^{2} \)
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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

−12.94425203941576679350960600445, −11.86761125991464495033899936367, −10.47071364672869733886264071206, −10.11165890453147929645815197318, −8.736549206105787465677857261855, −7.46771146121027346111215982241, −6.64602074035611539185754559942, −4.53375325825089450735562831994, −1.85610449504001790581175206707, −0.48499600577140579031276441423, 0.954892976883218232022068013791, 2.78601014358653631510779113100, 5.59882377527006706239450094541, 6.80097766330869808618157420801, 8.166543413809720024102668639030, 9.099964332350213715547823109990, 10.22430694735805833761680627208, 11.30115203695132746266717616498, 11.93738143199234512863210150541, 13.75410642181486189651452575566

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