Properties

Label 2-6-3.2-c24-0-0
Degree $2$
Conductor $6$
Sign $-0.383 - 0.923i$
Analytic cond. $21.8980$
Root an. cond. $4.67953$
Motivic weight $24$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.89e3i·2-s + (−4.90e5 + 2.03e5i)3-s − 8.38e6·4-s − 1.24e8i·5-s + (−5.90e8 − 1.42e9i)6-s + 1.50e10·7-s − 2.42e10i·8-s + (1.99e11 − 2.00e11i)9-s + 3.59e11·10-s + 2.25e12i·11-s + (4.11e12 − 1.70e12i)12-s + 4.78e12·13-s + 4.36e13i·14-s + (2.52e13 + 6.08e13i)15-s + 7.03e13·16-s − 5.31e14i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.923 + 0.383i)3-s − 0.500·4-s − 0.507i·5-s + (−0.271 − 0.653i)6-s + 1.08·7-s − 0.353i·8-s + (0.705 − 0.708i)9-s + 0.359·10-s + 0.719i·11-s + (0.461 − 0.191i)12-s + 0.205·13-s + 0.769i·14-s + (0.194 + 0.469i)15-s + 0.250·16-s − 0.912i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(25-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+12) \, L(s)\cr =\mathstrut & (-0.383 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6\)    =    \(2 \cdot 3\)
Sign: $-0.383 - 0.923i$
Analytic conductor: \(21.8980\)
Root analytic conductor: \(4.67953\)
Motivic weight: \(24\)
Rational: no
Arithmetic: yes
Character: $\chi_{6} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 6,\ (\ :12),\ -0.383 - 0.923i)\)

Particular Values

\(L(\frac{25}{2})\) \(\approx\) \(0.714172 + 1.06974i\)
\(L(\frac12)\) \(\approx\) \(0.714172 + 1.06974i\)
\(L(13)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2.89e3iT \)
3 \( 1 + (4.90e5 - 2.03e5i)T \)
good5 \( 1 + 1.24e8iT - 5.96e16T^{2} \)
7 \( 1 - 1.50e10T + 1.91e20T^{2} \)
11 \( 1 - 2.25e12iT - 9.84e24T^{2} \)
13 \( 1 - 4.78e12T + 5.42e26T^{2} \)
17 \( 1 + 5.31e14iT - 3.39e29T^{2} \)
19 \( 1 + 2.04e15T + 4.89e30T^{2} \)
23 \( 1 - 3.16e16iT - 4.80e32T^{2} \)
29 \( 1 - 6.82e17iT - 1.25e35T^{2} \)
31 \( 1 + 6.83e17T + 6.20e35T^{2} \)
37 \( 1 - 9.74e18T + 4.33e37T^{2} \)
41 \( 1 + 1.91e19iT - 5.09e38T^{2} \)
43 \( 1 - 6.07e19T + 1.59e39T^{2} \)
47 \( 1 - 1.14e20iT - 1.35e40T^{2} \)
53 \( 1 - 5.21e20iT - 2.41e41T^{2} \)
59 \( 1 - 5.99e20iT - 3.16e42T^{2} \)
61 \( 1 - 7.42e19T + 7.04e42T^{2} \)
67 \( 1 - 8.52e21T + 6.69e43T^{2} \)
71 \( 1 - 2.05e22iT - 2.69e44T^{2} \)
73 \( 1 + 3.58e22T + 5.24e44T^{2} \)
79 \( 1 - 2.61e22T + 3.49e45T^{2} \)
83 \( 1 - 1.17e23iT - 1.14e46T^{2} \)
89 \( 1 + 1.60e23iT - 6.10e46T^{2} \)
97 \( 1 + 8.32e22T + 4.81e47T^{2} \)
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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

−17.25786551027797784600595997502, −16.00212979901184275627076525962, −14.63066364585977864343584052937, −12.62466226841399312815614632508, −11.02928660944804656581412298752, −9.168556175756802540282662337337, −7.29204491314306425291681881692, −5.43916590617675305517866381095, −4.41935538334447090486624688009, −1.18930130381941536553543523570, 0.61232541560928236478977761919, 2.12495071391108041714130029764, 4.39453258195576504780431382451, 6.13435039509716438315737053670, 8.168742792580433268235536785978, 10.59373396462902352122798932441, 11.39988092480318444467049449568, 12.94886168992554985012470801522, 14.63811515958004050938470050000, 16.84463310666481914551000674607

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