Properties

Label 2-4140-23.22-c2-0-61
Degree $2$
Conductor $4140$
Sign $-0.325 + 0.945i$
Analytic cond. $112.806$
Root an. cond. $10.6210$
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  + 2.23i·5-s − 6.51i·7-s − 16.5i·11-s − 12.7·13-s − 15.5i·17-s + 21.3i·19-s + (21.7 + 7.49i)23-s − 5.00·25-s + 21.6·29-s + 61.2·31-s + 14.5·35-s − 29.9i·37-s + 57.4·41-s + 61.2i·43-s − 72.3·47-s + ⋯
L(s)  = 1  + 0.447i·5-s − 0.930i·7-s − 1.50i·11-s − 0.977·13-s − 0.911i·17-s + 1.12i·19-s + (0.945 + 0.325i)23-s − 0.200·25-s + 0.745·29-s + 1.97·31-s + 0.416·35-s − 0.809i·37-s + 1.40·41-s + 1.42i·43-s − 1.54·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.325 + 0.945i$
Analytic conductor: \(112.806\)
Root analytic conductor: \(10.6210\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1),\ -0.325 + 0.945i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.626386649\)
\(L(\frac12)\) \(\approx\) \(1.626386649\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - 2.23iT \)
23 \( 1 + (-21.7 - 7.49i)T \)
good7 \( 1 + 6.51iT - 49T^{2} \)
11 \( 1 + 16.5iT - 121T^{2} \)
13 \( 1 + 12.7T + 169T^{2} \)
17 \( 1 + 15.5iT - 289T^{2} \)
19 \( 1 - 21.3iT - 361T^{2} \)
29 \( 1 - 21.6T + 841T^{2} \)
31 \( 1 - 61.2T + 961T^{2} \)
37 \( 1 + 29.9iT - 1.36e3T^{2} \)
41 \( 1 - 57.4T + 1.68e3T^{2} \)
43 \( 1 - 61.2iT - 1.84e3T^{2} \)
47 \( 1 + 72.3T + 2.20e3T^{2} \)
53 \( 1 - 98.7iT - 2.80e3T^{2} \)
59 \( 1 + 40.4T + 3.48e3T^{2} \)
61 \( 1 + 48.2iT - 3.72e3T^{2} \)
67 \( 1 + 70.9iT - 4.48e3T^{2} \)
71 \( 1 - 62.9T + 5.04e3T^{2} \)
73 \( 1 + 13.0T + 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 - 16.3iT - 6.88e3T^{2} \)
89 \( 1 + 101. iT - 7.92e3T^{2} \)
97 \( 1 + 148. iT - 9.40e3T^{2} \)
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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

−7.81464354921369882268641088564, −7.47187032191130241442814945517, −6.49238108419857409515633353553, −5.98823525271659958781576304410, −4.94086129944376874488307155531, −4.26458159014022063035439978147, −3.17382120450097936786042163159, −2.77897602511944965165339222489, −1.25164217426803848708086299727, −0.38724521894257905550988439705, 1.04567141624736071498211125387, 2.27144986497243395361235915653, 2.70181359396326684039042360465, 4.10615142060884924722289051831, 4.90126558747983606062445661909, 5.19531738423234178666879535842, 6.45875046038238889347182278351, 6.86976175761620494673187448108, 7.83781909551508953874553787063, 8.496572948960193095112652648999

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