Properties

Label 2-2352-12.11-c1-0-60
Degree $2$
Conductor $2352$
Sign $0.816 + 0.577i$
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  + (1.41 + i)3-s − 1.41i·5-s + (1.00 + 2.82i)9-s − 1.41·11-s − 2·13-s + (1.41 − 2.00i)15-s − 7.07i·17-s − 4i·19-s + 7.07·23-s + 2.99·25-s + (−1.41 + 5.00i)27-s − 2.82i·29-s − 6i·31-s + (−2.00 − 1.41i)33-s − 4·37-s + ⋯
L(s)  = 1  + (0.816 + 0.577i)3-s − 0.632i·5-s + (0.333 + 0.942i)9-s − 0.426·11-s − 0.554·13-s + (0.365 − 0.516i)15-s − 1.71i·17-s − 0.917i·19-s + 1.47·23-s + 0.599·25-s + (−0.272 + 0.962i)27-s − 0.525i·29-s − 1.07i·31-s + (−0.348 − 0.246i)33-s − 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.816 + 0.577i$
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: \((2,\ 2352,\ (\ :1/2),\ 0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.239895268\)
\(L(\frac12)\) \(\approx\) \(2.239895268\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 - i)T \)
7 \( 1 \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 7.07iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 - 7.07T + 23T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + 6iT - 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 - 7.07T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 - 9.89iT - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.965611139321906594475170989101, −8.334652334750396179107566055270, −7.37983061507115160597926706781, −6.91212149770500835134768259297, −5.28634358832790037908519631370, −5.00464343420923940090523869358, −4.14328079997480117371506743379, −2.93622319927902300706172653111, −2.39339501004370462916810343257, −0.73350901023005903068740718448, 1.28791014961678492540838418901, 2.33847562292306218577573793023, 3.20443249358079779490795880655, 3.93152627711715059196162055394, 5.16771072478647883491913355218, 6.11769899448783686891675218656, 6.99508093727146048488954288477, 7.37539308501466598486536425104, 8.444447160619701303934104825779, 8.739715908340123286415214347274

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