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Properties

Label 2-69-3.2-c2-0-3
Degree $2$
Conductor $69$
Sign $0.678 - 0.734i$
Analytic cond. $1.88011$
Root an. cond. $1.37117$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.29i·2-s + (−2.20 − 2.03i)3-s + 2.31·4-s + 5.31i·5-s + (2.64 − 2.86i)6-s + 12.0·7-s + 8.19i·8-s + (0.718 + 8.97i)9-s − 6.89·10-s − 11.3i·11-s + (−5.10 − 4.71i)12-s − 10.1·13-s + 15.6i·14-s + (10.8 − 11.7i)15-s − 1.36·16-s − 4.40i·17-s + ⋯
L(s)  = 1  + 0.648i·2-s + (−0.734 − 0.678i)3-s + 0.579·4-s + 1.06i·5-s + (0.440 − 0.476i)6-s + 1.71·7-s + 1.02i·8-s + (0.0798 + 0.996i)9-s − 0.689·10-s − 1.03i·11-s + (−0.425 − 0.392i)12-s − 0.779·13-s + 1.11i·14-s + (0.720 − 0.780i)15-s − 0.0854·16-s − 0.258i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(69\)    =    \(3 \cdot 23\)
Sign: $0.678 - 0.734i$
Analytic conductor: \(1.88011\)
Root analytic conductor: \(1.37117\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{69} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 69,\ (\ :1),\ 0.678 - 0.734i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.15635 + 0.506270i\)
\(L(\frac12)\) \(\approx\) \(1.15635 + 0.506270i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.20 + 2.03i)T \)
23 \( 1 + 4.79iT \)
good2 \( 1 - 1.29iT - 4T^{2} \)
5 \( 1 - 5.31iT - 25T^{2} \)
7 \( 1 - 12.0T + 49T^{2} \)
11 \( 1 + 11.3iT - 121T^{2} \)
13 \( 1 + 10.1T + 169T^{2} \)
17 \( 1 + 4.40iT - 289T^{2} \)
19 \( 1 + 14.8T + 361T^{2} \)
29 \( 1 + 33.0iT - 841T^{2} \)
31 \( 1 + 22.9T + 961T^{2} \)
37 \( 1 + 42.4T + 1.36e3T^{2} \)
41 \( 1 + 33.1iT - 1.68e3T^{2} \)
43 \( 1 - 16.0T + 1.84e3T^{2} \)
47 \( 1 - 74.0iT - 2.20e3T^{2} \)
53 \( 1 + 22.5iT - 2.80e3T^{2} \)
59 \( 1 + 96.2iT - 3.48e3T^{2} \)
61 \( 1 + 93.9T + 3.72e3T^{2} \)
67 \( 1 - 126.T + 4.48e3T^{2} \)
71 \( 1 + 65.3iT - 5.04e3T^{2} \)
73 \( 1 - 22.0T + 5.32e3T^{2} \)
79 \( 1 + 37.4T + 6.24e3T^{2} \)
83 \( 1 - 73.5iT - 6.88e3T^{2} \)
89 \( 1 - 72.9iT - 7.92e3T^{2} \)
97 \( 1 - 0.728T + 9.40e3T^{2} \)
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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

−14.54577372203426354691008748580, −14.00422309422518919443385515944, −12.15686829870793590243417169570, −11.09349893069955809187612861244, −10.83800049803167233055233198487, −8.214622161261227533955886520496, −7.41011670871996816480744230803, −6.33484512817050293705068378866, −5.14072250196282626190761151168, −2.21303242629458173859764704959, 1.63464243621635988174400551265, 4.35468980641529923293689222758, 5.26274209118153803820922542001, 7.20760682299593594317724354143, 8.800894231029519441526828394246, 10.15941652463104302838231787317, 11.07000041245280216261035341265, 12.05533215150293078716556195785, 12.63386440639310775875888354340, 14.66836127487628913244502210630

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