Properties

Label 2-2352-28.27-c1-0-17
Degree $2$
Conductor $2352$
Sign $0.933 - 0.358i$
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  + 3-s − 0.601i·5-s + 9-s + 0.163i·11-s + 3.37i·13-s − 0.601i·15-s − 2.13i·17-s + 0.953·19-s + 6.99i·23-s + 4.63·25-s + 27-s − 1.28·29-s + 3.35·31-s + 0.163i·33-s + 5.16·37-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.269i·5-s + 0.333·9-s + 0.0493i·11-s + 0.937i·13-s − 0.155i·15-s − 0.517i·17-s + 0.218·19-s + 1.45i·23-s + 0.927·25-s + 0.192·27-s − 0.239·29-s + 0.602·31-s + 0.0284i·33-s + 0.848·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.933 - 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.264129654\)
\(L(\frac12)\) \(\approx\) \(2.264129654\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 \)
good5 \( 1 + 0.601iT - 5T^{2} \)
11 \( 1 - 0.163iT - 11T^{2} \)
13 \( 1 - 3.37iT - 13T^{2} \)
17 \( 1 + 2.13iT - 17T^{2} \)
19 \( 1 - 0.953T + 19T^{2} \)
23 \( 1 - 6.99iT - 23T^{2} \)
29 \( 1 + 1.28T + 29T^{2} \)
31 \( 1 - 3.35T + 31T^{2} \)
37 \( 1 - 5.16T + 37T^{2} \)
41 \( 1 - 6.67iT - 41T^{2} \)
43 \( 1 + 6.09iT - 43T^{2} \)
47 \( 1 + 3.69T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 - 8.05T + 59T^{2} \)
61 \( 1 - 0.181iT - 61T^{2} \)
67 \( 1 - 9.09iT - 67T^{2} \)
71 \( 1 - 3.36iT - 71T^{2} \)
73 \( 1 + 12.1iT - 73T^{2} \)
79 \( 1 - 6.98iT - 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 14.8iT - 89T^{2} \)
97 \( 1 - 9.08iT - 97T^{2} \)
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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

−8.989116838824046740593173398754, −8.402116995787331893767117370755, −7.42379777813853792056711700827, −6.94005473462300694097089038776, −5.88406194599786992179199162536, −4.96818067716932708022111172223, −4.17136606098289181855480017969, −3.24963041328052264143719738744, −2.25078439029949990646889721133, −1.12954921798893029611466155470, 0.855722908512736566870775507038, 2.31505538536968367248004785318, 3.04567783545437821169989824999, 3.99644063229530882532854614947, 4.90117792699665666875506268388, 5.87523544741277643540835792249, 6.69168700707592570941253504031, 7.46505941017333924986247508026, 8.312578932623884200112060387089, 8.722147523588057683775111441431

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