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Properties

Label 2-2380-595.489-c0-0-0
Degree $2$
Conductor $2380$
Sign $-0.602 - 0.798i$
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  + (1.22 + 1.22i)3-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.99i·9-s + (−0.366 + 0.366i)11-s − 0.517·13-s − 1.73i·15-s + (−0.258 + 0.965i)17-s − 1.73·21-s + 1.00i·25-s + (−1.22 + 1.22i)27-s + (1.36 + 1.36i)29-s − 0.896·33-s + 1.00·35-s + (−0.633 − 0.633i)39-s + ⋯
L(s)  = 1  + (1.22 + 1.22i)3-s + (−0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.99i·9-s + (−0.366 + 0.366i)11-s − 0.517·13-s − 1.73i·15-s + (−0.258 + 0.965i)17-s − 1.73·21-s + 1.00i·25-s + (−1.22 + 1.22i)27-s + (1.36 + 1.36i)29-s − 0.896·33-s + 1.00·35-s + (−0.633 − 0.633i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.602 - 0.798i$
Analytic conductor: \(1.18777\)
Root analytic conductor: \(1.08985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ -0.602 - 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.220068110\)
\(L(\frac12)\) \(\approx\) \(1.220068110\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (0.258 - 0.965i)T \)
good3 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
11 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
13 \( 1 + 0.517T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + 1.93T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1 + i)T + iT^{2} \)
73 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
79 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
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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

−9.280765120622344010982960741123, −8.705558591939372249831419205939, −8.250939721494040909754329694512, −7.42468263465204250976993782884, −6.29927345186318462667179012646, −5.01992410232157990626405986980, −4.69146460852683310718788451745, −3.61661564296060043067415754270, −3.08582748298098975538635347689, −2.00600723895724675878598053835, 0.68180782603531714250485271104, 2.31815070290462142782795831097, 2.96357815618201921501448455663, 3.63151529668490785712575685864, 4.70645458338054150924543313148, 6.31297079397253671239987471371, 6.73509858330711399346907284098, 7.48505855301829413399456501429, 7.912846707495171074674490569523, 8.628632966070100157274209530172

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