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Properties

Label 2-2380-2380.1427-c0-0-7
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 + (0.156 + 0.987i)5-s + (−0.863 + 1.69i)6-s + (−0.707 + 0.707i)7-s + (0.809 + 0.587i)8-s + 2.61i·9-s + (0.891 − 0.453i)10-s + (1.87 + 0.297i)12-s + (0.891 + 0.453i)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 + (0.156 + 0.987i)5-s + (−0.863 + 1.69i)6-s + (−0.707 + 0.707i)7-s + (0.809 + 0.587i)8-s + 2.61i·9-s + (0.891 − 0.453i)10-s + (1.87 + 0.297i)12-s + (0.891 + 0.453i)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

\(L(\frac{1}{2})\) \(\approx\) \(0.2200652213\)
\(L(\frac12)\) \(\approx\) \(0.2200652213\)
\(L(1)\) 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 \( 1 + (-0.156 - 0.987i)T \)
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 \( 1 + 0.312iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.78T + T^{2} \)
43 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.907T + T^{2} \)
67 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
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

−8.738857189621636020559089689011, −7.928017037324063090032099301560, −6.86912976667406812039086057403, −6.73975282045325316029294181393, −5.61674115188759642957080279455, −5.01575962039635286726347578309, −3.54403930598806076523956462998, −2.46162824287803306284612647781, −1.87824258015921376691261008820, −0.23022992711889408612598310086, 1.10361651391728393228813074795, 3.67594659039608820880894761869, 4.26858029627544478616356087406, 4.97562962635618265062319733532, 5.60394008556861771646155943609, 6.40092771695290809750315986596, 6.89282810429118017114373461100, 8.220222783337443902032895290612, 8.932919796769556324563763218252, 9.635080858260541200270606562344

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