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Properties

Label 2-4608-16.13-c1-0-45
Degree $2$
Conductor $4608$
Sign $0.382 + 0.923i$
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  + (−2.23 + 2.23i)5-s − 4.57i·7-s + (−1.74 + 1.74i)11-s + (1 + i)13-s − 6.47·17-s + (1.74 + 1.74i)19-s + 5.65i·23-s − 5.00i·25-s + (0.236 + 0.236i)29-s + 10.2·31-s + (10.2 + 10.2i)35-s + (−1.47 + 1.47i)37-s − 6.47i·41-s + (−7.40 + 7.40i)43-s − 13.9·49-s + ⋯
L(s)  = 1  + (−0.999 + 0.999i)5-s − 1.72i·7-s + (−0.527 + 0.527i)11-s + (0.277 + 0.277i)13-s − 1.56·17-s + (0.401 + 0.401i)19-s + 1.17i·23-s − 1.00i·25-s + (0.0438 + 0.0438i)29-s + 1.83·31-s + (1.72 + 1.72i)35-s + (−0.242 + 0.242i)37-s − 1.01i·41-s + (−1.12 + 1.12i)43-s − 1.99·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4608\)    =    \(2^{9} \cdot 3^{2}\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(36.7950\)
Root analytic conductor: \(6.06589\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4608} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4608,\ (\ :1/2),\ 0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8636353711\)
\(L(\frac12)\) \(\approx\) \(0.8636353711\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 2.23i)T - 5iT^{2} \)
7 \( 1 + 4.57iT - 7T^{2} \)
11 \( 1 + (1.74 - 1.74i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 6.47T + 17T^{2} \)
19 \( 1 + (-1.74 - 1.74i)T + 19iT^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + (-0.236 - 0.236i)T + 29iT^{2} \)
31 \( 1 - 10.2T + 31T^{2} \)
37 \( 1 + (1.47 - 1.47i)T - 37iT^{2} \)
41 \( 1 + 6.47iT - 41T^{2} \)
43 \( 1 + (7.40 - 7.40i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-3.76 + 3.76i)T - 53iT^{2} \)
59 \( 1 + (-6.32 + 6.32i)T - 59iT^{2} \)
61 \( 1 + (-7.47 - 7.47i)T + 61iT^{2} \)
67 \( 1 + (8.48 + 8.48i)T + 67iT^{2} \)
71 \( 1 + 3.49iT - 71T^{2} \)
73 \( 1 + 14.9iT - 73T^{2} \)
79 \( 1 + 1.08T + 79T^{2} \)
83 \( 1 + (1.74 + 1.74i)T + 83iT^{2} \)
89 \( 1 + 10iT - 89T^{2} \)
97 \( 1 - 4.94T + 97T^{2} \)
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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

−7.86581052477755250151557062239, −7.48555013848552876998120533427, −6.82716202608116277030260032478, −6.38823788590941055483592736833, −4.96728399513128851497024211656, −4.28525577636935316016294426576, −3.68569732996019339449169449197, −2.94983043080666218199127587640, −1.69055145384047169342843646587, −0.32233182859220640735170375509, 0.811103565947265335942213973914, 2.31411655084975724817799231177, 2.89454623819484194010072713512, 4.09345808717805372986012352420, 4.80891128245670763923788377821, 5.38442909479499956657515620436, 6.21424349186892414658845741060, 6.96150483269729371087019387042, 8.158322632551244536372217120130, 8.570327335477391021445130966692

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