Properties

Label 2-2352-1.1-c3-0-63
Degree $2$
Conductor $2352$
Sign $1$
Analytic cond. $138.772$
Root an. cond. $11.7801$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 17.8·5-s + 9·9-s + 11.3·11-s − 13.0·13-s + 53.6·15-s + 53.2·17-s + 42.4·19-s − 152.·23-s + 194.·25-s + 27·27-s + 186.·29-s + 157.·31-s + 34.1·33-s + 3.74·37-s − 39.2·39-s − 39.3·41-s − 429.·43-s + 160.·45-s − 21.1·47-s + 159.·51-s + 365.·53-s + 203.·55-s + 127.·57-s + 226.·59-s + 651.·61-s − 234.·65-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.59·5-s + 0.333·9-s + 0.312·11-s − 0.279·13-s + 0.922·15-s + 0.759·17-s + 0.512·19-s − 1.37·23-s + 1.55·25-s + 0.192·27-s + 1.19·29-s + 0.914·31-s + 0.180·33-s + 0.0166·37-s − 0.161·39-s − 0.149·41-s − 1.52·43-s + 0.532·45-s − 0.0657·47-s + 0.438·51-s + 0.948·53-s + 0.499·55-s + 0.295·57-s + 0.499·59-s + 1.36·61-s − 0.446·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(138.772\)
Root analytic conductor: \(11.7801\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.503515379\)
\(L(\frac12)\) \(\approx\) \(4.503515379\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 \)
good5 \( 1 - 17.8T + 125T^{2} \)
11 \( 1 - 11.3T + 1.33e3T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 - 53.2T + 4.91e3T^{2} \)
19 \( 1 - 42.4T + 6.85e3T^{2} \)
23 \( 1 + 152.T + 1.21e4T^{2} \)
29 \( 1 - 186.T + 2.43e4T^{2} \)
31 \( 1 - 157.T + 2.97e4T^{2} \)
37 \( 1 - 3.74T + 5.06e4T^{2} \)
41 \( 1 + 39.3T + 6.89e4T^{2} \)
43 \( 1 + 429.T + 7.95e4T^{2} \)
47 \( 1 + 21.1T + 1.03e5T^{2} \)
53 \( 1 - 365.T + 1.48e5T^{2} \)
59 \( 1 - 226.T + 2.05e5T^{2} \)
61 \( 1 - 651.T + 2.26e5T^{2} \)
67 \( 1 + 145.T + 3.00e5T^{2} \)
71 \( 1 - 368.T + 3.57e5T^{2} \)
73 \( 1 - 608.T + 3.89e5T^{2} \)
79 \( 1 + 910.T + 4.93e5T^{2} \)
83 \( 1 - 327.T + 5.71e5T^{2} \)
89 \( 1 + 37.6T + 7.04e5T^{2} \)
97 \( 1 - 722.T + 9.12e5T^{2} \)
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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

−8.636002510099461980670447650721, −8.044724256048358869499289151341, −6.97612083470199563298865907453, −6.30320346918028220602283856005, −5.54547036376210995587926514553, −4.76098081866589802598218651933, −3.61333664914902805674795878980, −2.64741704376159549904435017706, −1.90147094551224845955926339891, −0.963459406538530712534210879917, 0.963459406538530712534210879917, 1.90147094551224845955926339891, 2.64741704376159549904435017706, 3.61333664914902805674795878980, 4.76098081866589802598218651933, 5.54547036376210995587926514553, 6.30320346918028220602283856005, 6.97612083470199563298865907453, 8.044724256048358869499289151341, 8.636002510099461980670447650721

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