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Properties

Label 2-4788-133.132-c1-0-0
Degree $2$
Conductor $4788$
Sign $-0.826 + 0.562i$
Analytic cond. $38.2323$
Root an. cond. $6.18323$
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  + 3.31i·5-s + (1.88 − 1.86i)7-s + 4.03·11-s − 4.35·13-s + 3.96i·17-s + (−4.28 − 0.793i)19-s − 5.24·23-s − 5.98·25-s − 6.11i·29-s − 5.77·31-s + (6.16 + 6.23i)35-s − 4.33i·37-s − 6.66·41-s − 6.98·43-s − 8.19i·47-s + ⋯
L(s)  = 1  + 1.48i·5-s + (0.710 − 0.703i)7-s + 1.21·11-s − 1.20·13-s + 0.961i·17-s + (−0.983 − 0.181i)19-s − 1.09·23-s − 1.19·25-s − 1.13i·29-s − 1.03·31-s + (1.04 + 1.05i)35-s − 0.712i·37-s − 1.04·41-s − 1.06·43-s − 1.19i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $-0.826 + 0.562i$
Analytic conductor: \(38.2323\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4788} (3457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4788,\ (\ :1/2),\ -0.826 + 0.562i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.009964602554\)
\(L(\frac12)\) \(\approx\) \(0.009964602554\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-1.88 + 1.86i)T \)
19 \( 1 + (4.28 + 0.793i)T \)
good5 \( 1 - 3.31iT - 5T^{2} \)
11 \( 1 - 4.03T + 11T^{2} \)
13 \( 1 + 4.35T + 13T^{2} \)
17 \( 1 - 3.96iT - 17T^{2} \)
23 \( 1 + 5.24T + 23T^{2} \)
29 \( 1 + 6.11iT - 29T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + 4.33iT - 37T^{2} \)
41 \( 1 + 6.66T + 41T^{2} \)
43 \( 1 + 6.98T + 43T^{2} \)
47 \( 1 + 8.19iT - 47T^{2} \)
53 \( 1 + 1.52iT - 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 1.72iT - 61T^{2} \)
67 \( 1 - 3.12iT - 67T^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 + 7.60iT - 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + 2.51iT - 83T^{2} \)
89 \( 1 - 3.82T + 89T^{2} \)
97 \( 1 - 9.44T + 97T^{2} \)
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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

−8.579147503588355217237112573070, −7.84420964657024732931187000855, −7.21058914767421447192476923322, −6.61716066003657443192061935400, −6.07158012240851586660266818983, −4.95814968537616113153886241039, −3.97957372503054585190705771757, −3.65271844411044495871755570160, −2.33635484348221369797337263827, −1.75633642539678826748989164876, 0.00247288249970185984369624553, 1.42935395758877046022502778032, 2.00333622235385667403420735051, 3.27978324258247517826771053869, 4.49127593494802835231548454676, 4.74484169100648762140912596651, 5.49930968233940864495998209963, 6.31685489659588789498328267873, 7.22477836381652413498697684967, 8.032676788145930042343540337772

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