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Properties

Label 2-2352-84.83-c0-0-1
Degree $2$
Conductor $2352$
Sign $0.944 - 0.327i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
Motivic weight $0$
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 + 9-s + 1.73i·13-s + 19-s − 25-s + 27-s − 31-s + 37-s + 1.73i·39-s − 1.73i·43-s + 57-s − 1.73i·67-s + 1.73i·73-s − 75-s − 1.73i·79-s + ⋯
L(s)  = 1  + 3-s + 9-s + 1.73i·13-s + 19-s − 25-s + 27-s − 31-s + 37-s + 1.73i·39-s − 1.73i·43-s + 57-s − 1.73i·67-s + 1.73i·73-s − 75-s − 1.73i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.738687825\)
\(L(\frac12)\) \(\approx\) \(1.738687825\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.73iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.290482101164245452291464194591, −8.542173734587199270846370966433, −7.62559400836656336361540568784, −7.12015646167438576960753738447, −6.24842367342337009154460249183, −5.14581485904097073347013384649, −4.16479207280695905422706500178, −3.57739591976688341342433447908, −2.39580999772606537566726123613, −1.58309807623782885716503054670, 1.24286079441251207481466677066, 2.56947549379523153126500478508, 3.25750276778918626683902306310, 4.11459919652790620964795266808, 5.19698848139289390430207371670, 5.94431376429737364626557300661, 7.06365802641872893874730740151, 7.88251214804175251153751013057, 8.093992730379442651950593693811, 9.222062084050737052921673004828

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