Properties

Label 2-2352-12.11-c1-0-76
Degree $2$
Conductor $2352$
Sign $-0.305 + 0.952i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (1.69 − 0.366i)3-s − 3.38i·5-s + (2.73 − 1.23i)9-s − 2.47·11-s + 6.19·13-s + (−1.23 − 5.73i)15-s − 2.47i·17-s − 0.732i·19-s − 6.77·23-s − 6.46·25-s + (4.17 − 3.09i)27-s − 2.47i·29-s − 9.46i·31-s + (−4.19 + 0.907i)33-s + 4.53·37-s + ⋯
L(s)  = 1  + (0.977 − 0.211i)3-s − 1.51i·5-s + (0.910 − 0.413i)9-s − 0.747·11-s + 1.71·13-s + (−0.319 − 1.48i)15-s − 0.601i·17-s − 0.167i·19-s − 1.41·23-s − 1.29·25-s + (0.802 − 0.596i)27-s − 0.460i·29-s − 1.69i·31-s + (−0.730 + 0.157i)33-s + 0.745·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.305 + 0.952i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.512361067\)
\(L(\frac12)\) \(\approx\) \(2.512361067\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 + 0.366i)T \)
7 \( 1 \)
good5 \( 1 + 3.38iT - 5T^{2} \)
11 \( 1 + 2.47T + 11T^{2} \)
13 \( 1 - 6.19T + 13T^{2} \)
17 \( 1 + 2.47iT - 17T^{2} \)
19 \( 1 + 0.732iT - 19T^{2} \)
23 \( 1 + 6.77T + 23T^{2} \)
29 \( 1 + 2.47iT - 29T^{2} \)
31 \( 1 + 9.46iT - 31T^{2} \)
37 \( 1 - 4.53T + 37T^{2} \)
41 \( 1 - 9.25iT - 41T^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9.25iT - 53T^{2} \)
59 \( 1 + 8.34T + 59T^{2} \)
61 \( 1 - 4.73T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 9.25T + 71T^{2} \)
73 \( 1 - 4.53T + 73T^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + 8.34T + 83T^{2} \)
89 \( 1 + 14.2iT - 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−8.569033293168988108723059255913, −8.146598803503209878141143718879, −7.63006236187559040431556002989, −6.30964602756757116569213943768, −5.66208260410227889781817129945, −4.49351901008994897540884051953, −4.02850531843750363901352751563, −2.84833624343232621892460557127, −1.75050319048958225446885607624, −0.75932729113819855446793362125, 1.68921469342184227922444795239, 2.62553257322721061520433863406, 3.52666921036854123447430467798, 3.91480076508314564482236145217, 5.33774060937932795141308645489, 6.32263139146159237515694666593, 6.89637017483316612476296619768, 7.85362267538698299537433927258, 8.313540958460296968589361455135, 9.131883765619978660124504396057

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