Properties

Label 2-4608-12.11-c1-0-18
Degree $2$
Conductor $4608$
Sign $-0.577 - 0.816i$
Analytic cond. $36.7950$
Root an. cond. $6.06589$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.82i·5-s + 1.41i·7-s + 2·13-s + 1.41i·17-s + 2.82i·19-s + 6·23-s − 3.00·25-s − 5.65i·29-s + 9.89i·31-s − 4.00·35-s − 2·37-s − 4.24i·41-s + 8.48i·43-s + 10·47-s + 5·49-s + ⋯
L(s)  = 1  + 1.26i·5-s + 0.534i·7-s + 0.554·13-s + 0.342i·17-s + 0.648i·19-s + 1.25·23-s − 0.600·25-s − 1.05i·29-s + 1.77i·31-s − 0.676·35-s − 0.328·37-s − 0.662i·41-s + 1.29i·43-s + 1.45·47-s + 0.714·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (4607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.804818769\)
\(L(\frac12)\) \(\approx\) \(1.804818769\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 - 2.82iT - 5T^{2} \)
7 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 5.65iT - 29T^{2} \)
31 \( 1 - 9.89iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 - 8.48iT - 43T^{2} \)
47 \( 1 - 10T + 47T^{2} \)
53 \( 1 - 5.65iT - 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 5.65iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + 10T + 73T^{2} \)
79 \( 1 + 12.7iT - 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 9.89iT - 89T^{2} \)
97 \( 1 - 12T + 97T^{2} \)
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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

−8.708939325123522326294394013957, −7.69000874155822862744733328026, −7.14378782845067483818111445199, −6.33330154045577279542136862733, −5.86655015292593048227883369808, −4.91944239709002797886671611082, −3.88778964315032066822358969136, −3.12083841645615103625805546097, −2.46825761757428208425754601203, −1.30307726128232653728848458857, 0.55307771627804534081808136931, 1.30562994308230449742493869779, 2.53062676954705139837227007184, 3.63990661946272883795797310598, 4.37908441592510775091517185739, 5.09075383026184885821874152957, 5.68564825947825612859627414756, 6.73455857713300195719825659145, 7.34189631075212151148084317252, 8.151184771621422146874822353018

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