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Properties

Label 2-69-23.22-c6-0-11
Degree $2$
Conductor $69$
Sign $0.990 - 0.139i$
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  − 0.368·2-s + 15.5·3-s − 63.8·4-s + 113. i·5-s − 5.74·6-s − 569. i·7-s + 47.1·8-s + 243·9-s − 41.9i·10-s + 814. i·11-s − 995.·12-s + 3.51e3·13-s + 210. i·14-s + 1.77e3i·15-s + 4.06e3·16-s + 7.48e3i·17-s + ⋯
L(s)  = 1  − 0.0460·2-s + 0.577·3-s − 0.997·4-s + 0.911i·5-s − 0.0265·6-s − 1.66i·7-s + 0.0920·8-s + 0.333·9-s − 0.0419i·10-s + 0.611i·11-s − 0.576·12-s + 1.59·13-s + 0.0765i·14-s + 0.526i·15-s + 0.993·16-s + 1.52i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.990 - 0.139i$
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: \((2,\ 69,\ (\ :3),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.86608 + 0.130762i\)
\(L(\frac12)\) \(\approx\) \(1.86608 + 0.130762i\)
\(L(4)\) 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 \( 1 + (-1.20e4 + 1.69e3i)T \)
good2 \( 1 + 0.368T + 64T^{2} \)
5 \( 1 - 113. iT - 1.56e4T^{2} \)
7 \( 1 + 569. iT - 1.17e5T^{2} \)
11 \( 1 - 814. iT - 1.77e6T^{2} \)
13 \( 1 - 3.51e3T + 4.82e6T^{2} \)
17 \( 1 - 7.48e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.28e3iT - 4.70e7T^{2} \)
29 \( 1 + 3.31e3T + 5.94e8T^{2} \)
31 \( 1 - 3.44e4T + 8.87e8T^{2} \)
37 \( 1 - 7.16e3iT - 2.56e9T^{2} \)
41 \( 1 - 7.79e4T + 4.75e9T^{2} \)
43 \( 1 + 1.13e5iT - 6.32e9T^{2} \)
47 \( 1 + 6.28e4T + 1.07e10T^{2} \)
53 \( 1 - 1.35e5iT - 2.21e10T^{2} \)
59 \( 1 + 1.01e5T + 4.21e10T^{2} \)
61 \( 1 - 3.82e4iT - 5.15e10T^{2} \)
67 \( 1 + 2.29e5iT - 9.04e10T^{2} \)
71 \( 1 + 6.45e4T + 1.28e11T^{2} \)
73 \( 1 + 5.36e4T + 1.51e11T^{2} \)
79 \( 1 - 4.58e5iT - 2.43e11T^{2} \)
83 \( 1 + 1.13e6iT - 3.26e11T^{2} \)
89 \( 1 - 8.29e5iT - 4.96e11T^{2} \)
97 \( 1 - 5.25e5iT - 8.32e11T^{2} \)
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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

−13.60188363155945853858605089797, −12.94292424947420145556283316456, −10.82221458207761763435979736509, −10.29865493061060847097298751135, −8.880658512597973730503718317109, −7.71922113122208140140906365855, −6.52571123632975216834463886364, −4.35292891145870406726161929510, −3.45854959055101569248835641029, −1.08776659330188788069541314758, 1.03739313377051500381363339996, 3.10560147546748576960346051071, 4.79166797059639716483985096452, 5.88908536842965233169382111671, 8.240645063229231477909961993707, 8.826152967916648799482275838998, 9.509807089755636975561949154071, 11.43842285772682047325618599049, 12.65821620025433912565860795369, 13.40662244386811736842564163938

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