Properties

Degree $2$
Conductor $2352$
Sign $0.605 + 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)3-s + (1 + 1.73i)5-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 4·13-s − 1.99·15-s + (3 − 5.19i)17-s + (−4 − 6.92i)19-s + (−3 − 5.19i)23-s + (0.500 − 0.866i)25-s + 0.999·27-s − 10·29-s + (−2 + 3.46i)31-s + (0.999 + 1.73i)33-s + (−3 − 5.19i)37-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 1.10·13-s − 0.516·15-s + (0.727 − 1.26i)17-s + (−0.917 − 1.58i)19-s + (−0.625 − 1.08i)23-s + (0.100 − 0.173i)25-s + 0.192·27-s − 1.85·29-s + (−0.359 + 0.622i)31-s + (0.174 + 0.301i)33-s + (−0.493 − 0.854i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Motivic weight: \(1\)
Character: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.405903075\)
\(L(\frac12)\) \(\approx\) \(1.405903075\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (4 + 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1 - 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 + (-2 + 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4T + 97T^{2} \)
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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

−9.026924650502198397311794069882, −8.238293653941586042831929169610, −7.08913600098806334463234592185, −6.53097058987306716963352900002, −5.78436641554897193390401523661, −4.96992830920037866504649465723, −3.92702906281358700852487197379, −3.13378328013269600317980961030, −2.14189477527070864479060436619, −0.49420185314976220282805971217, 1.50404947369174956930445069323, 1.72401889027666033798574163309, 3.56394001544549904949620868212, 4.16078704028136684001010309938, 5.52085651025831516810362680221, 5.82846772557277040098980208945, 6.60911955736818105125185209772, 7.82873433281682297105215498016, 8.118968934213075201095250999136, 9.124295105806833773727832379308

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