Properties

Label 2-69-23.22-c6-0-23
Degree $2$
Conductor $69$
Sign $0.777 + 0.628i$
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  + 13.5·2-s + 15.5·3-s + 119.·4-s − 141. i·5-s + 211.·6-s − 238. i·7-s + 749.·8-s + 243·9-s − 1.91e3i·10-s + 253. i·11-s + 1.86e3·12-s + 572.·13-s − 3.23e3i·14-s − 2.19e3i·15-s + 2.50e3·16-s + 197. i·17-s + ⋯
L(s)  = 1  + 1.69·2-s + 0.577·3-s + 1.86·4-s − 1.12i·5-s + 0.977·6-s − 0.696i·7-s + 1.46·8-s + 0.333·9-s − 1.91i·10-s + 0.190i·11-s + 1.07·12-s + 0.260·13-s − 1.17i·14-s − 0.651i·15-s + 0.612·16-s + 0.0402i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.777 + 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.777 + 0.628i$
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: \((2,\ 69,\ (\ :3),\ 0.777 + 0.628i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(5.37405 - 1.89962i\)
\(L(\frac12)\) \(\approx\) \(5.37405 - 1.89962i\)
\(L(4)\) 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 \( 1 + (-9.46e3 - 7.64e3i)T \)
good2 \( 1 - 13.5T + 64T^{2} \)
5 \( 1 + 141. iT - 1.56e4T^{2} \)
7 \( 1 + 238. iT - 1.17e5T^{2} \)
11 \( 1 - 253. iT - 1.77e6T^{2} \)
13 \( 1 - 572.T + 4.82e6T^{2} \)
17 \( 1 - 197. iT - 2.41e7T^{2} \)
19 \( 1 - 1.21e4iT - 4.70e7T^{2} \)
29 \( 1 + 6.50e3T + 5.94e8T^{2} \)
31 \( 1 - 4.46e3T + 8.87e8T^{2} \)
37 \( 1 - 4.48e4iT - 2.56e9T^{2} \)
41 \( 1 + 5.65e4T + 4.75e9T^{2} \)
43 \( 1 + 1.19e5iT - 6.32e9T^{2} \)
47 \( 1 + 6.06e4T + 1.07e10T^{2} \)
53 \( 1 - 1.00e4iT - 2.21e10T^{2} \)
59 \( 1 - 4.84e4T + 4.21e10T^{2} \)
61 \( 1 + 1.69e5iT - 5.15e10T^{2} \)
67 \( 1 - 5.51e5iT - 9.04e10T^{2} \)
71 \( 1 + 2.56e5T + 1.28e11T^{2} \)
73 \( 1 + 7.23e5T + 1.51e11T^{2} \)
79 \( 1 - 4.30e5iT - 2.43e11T^{2} \)
83 \( 1 - 9.00e4iT - 3.26e11T^{2} \)
89 \( 1 + 8.68e5iT - 4.96e11T^{2} \)
97 \( 1 + 1.17e6iT - 8.32e11T^{2} \)
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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

−13.38380636133298489676269376850, −12.69051308006555939248213891541, −11.70997080877629216734820440470, −10.16679587403933645072137932286, −8.583388078798006461954875952901, −7.19073939133684497737689015507, −5.64074488232798225928084881228, −4.46337739101797154455678940786, −3.45351097539944370831891594219, −1.54043034270718391783293936165, 2.46949990820841808033922840827, 3.23319709561231308962916785480, 4.78519440965354779046868281883, 6.24459617557415929701743047472, 7.18771240367531644712035251271, 8.985367343542106695061175869400, 10.75533761987536002464265341078, 11.60602874391799285233013562305, 12.88839710144175231456671498579, 13.67955752254842433656284723259

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