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Properties

Label 2-2380-2380.1427-c0-0-5
Degree $2$
Conductor $2380$
Sign $-0.973 - 0.229i$
Analytic cond. $1.18777$
Root an. cond. $1.08985$
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  + (−0.309 + 0.951i)2-s + (1.34 + 1.34i)3-s + (−0.809 − 0.587i)4-s + (0.987 + 0.156i)5-s + (−1.69 + 0.863i)6-s + (−0.707 + 0.707i)7-s + (0.809 − 0.587i)8-s + 2.61i·9-s + (−0.453 + 0.891i)10-s + (−0.297 − 1.87i)12-s + (−0.453 − 0.891i)14-s + (1.11 + 1.53i)15-s + (0.309 + 0.951i)16-s + (−0.707 − 0.707i)17-s + (−2.48 − 0.809i)18-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (1.34 + 1.34i)3-s + (−0.809 − 0.587i)4-s + (0.987 + 0.156i)5-s + (−1.69 + 0.863i)6-s + (−0.707 + 0.707i)7-s + (0.809 − 0.587i)8-s + 2.61i·9-s + (−0.453 + 0.891i)10-s + (−0.297 − 1.87i)12-s + (−0.453 − 0.891i)14-s + (1.11 + 1.53i)15-s + (0.309 + 0.951i)16-s + (−0.707 − 0.707i)17-s + (−2.48 − 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(1.18777\)
Root analytic conductor: \(1.08985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (1427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594659119\)
\(L(\frac12)\) \(\approx\) \(1.594659119\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.987 - 0.156i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
good3 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.97iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 0.907T + T^{2} \)
43 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.78T + T^{2} \)
67 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.34 - 1.34i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.831 - 0.831i)T + iT^{2} \)
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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

−9.353058643432715797336227819681, −9.029999892038229938231654442666, −8.177060199846275137823861994625, −7.39119047437029434841846104666, −6.32884515199826340182159840295, −5.61618013109713395293578138826, −4.80098936349360220639770229157, −4.00664045902728485141363286328, −2.88751553767092953182936254826, −2.15590566452575610219066717288, 1.08491514305354580980120844824, 1.88926848967920869298066708663, 2.74959060061627368377209498401, 3.44032762102994986961845047696, 4.42465249836147888554673514290, 5.90137119968696210447353778576, 6.77177779749748038255385876694, 7.35629333744845668600389747030, 8.264728943273299603500401808247, 8.951731069731522439917019916409

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