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Properties

Label 2-2385-265.239-c0-0-1
Degree $2$
Conductor $2385$
Sign $0.997 - 0.0759i$
Analytic cond. $1.19027$
Root an. cond. $1.09099$
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  + (0.0288 + 0.477i)2-s + (0.765 − 0.0929i)4-s + (0.297 − 0.954i)5-s + (0.152 + 0.833i)8-s + (0.464 + 0.114i)10-s + (0.354 − 0.0874i)16-s + (−0.587 − 0.851i)17-s + (1.12 + 1.43i)19-s + (0.138 − 0.758i)20-s + (0.501 − 0.501i)23-s + (−0.822 − 0.568i)25-s + (−0.0495 − 0.110i)31-s + (0.304 + 0.976i)32-s + (0.389 − 0.305i)34-s + (−0.653 + 0.578i)38-s + ⋯
L(s)  = 1  + (0.0288 + 0.477i)2-s + (0.765 − 0.0929i)4-s + (0.297 − 0.954i)5-s + (0.152 + 0.833i)8-s + (0.464 + 0.114i)10-s + (0.354 − 0.0874i)16-s + (−0.587 − 0.851i)17-s + (1.12 + 1.43i)19-s + (0.138 − 0.758i)20-s + (0.501 − 0.501i)23-s + (−0.822 − 0.568i)25-s + (−0.0495 − 0.110i)31-s + (0.304 + 0.976i)32-s + (0.389 − 0.305i)34-s + (−0.653 + 0.578i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2385\)    =    \(3^{2} \cdot 5 \cdot 53\)
Sign: $0.997 - 0.0759i$
Analytic conductor: \(1.19027\)
Root analytic conductor: \(1.09099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2385} (2359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2385,\ (\ :0),\ 0.997 - 0.0759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.614870438\)
\(L(\frac12)\) \(\approx\) \(1.614870438\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.297 + 0.954i)T \)
53 \( 1 + (-0.180 + 0.983i)T \)
good2 \( 1 + (-0.0288 - 0.477i)T + (-0.992 + 0.120i)T^{2} \)
7 \( 1 + (0.120 + 0.992i)T^{2} \)
11 \( 1 + (0.748 - 0.663i)T^{2} \)
13 \( 1 + (0.970 + 0.239i)T^{2} \)
17 \( 1 + (0.587 + 0.851i)T + (-0.354 + 0.935i)T^{2} \)
19 \( 1 + (-1.12 - 1.43i)T + (-0.239 + 0.970i)T^{2} \)
23 \( 1 + (-0.501 + 0.501i)T - iT^{2} \)
29 \( 1 + (0.748 + 0.663i)T^{2} \)
31 \( 1 + (0.0495 + 0.110i)T + (-0.663 + 0.748i)T^{2} \)
37 \( 1 + (0.885 - 0.464i)T^{2} \)
41 \( 1 + (0.663 + 0.748i)T^{2} \)
43 \( 1 + (0.885 + 0.464i)T^{2} \)
47 \( 1 + (0.731 - 1.39i)T + (-0.568 - 0.822i)T^{2} \)
59 \( 1 + (-0.568 - 0.822i)T^{2} \)
61 \( 1 + (-0.354 + 0.0649i)T + (0.935 - 0.354i)T^{2} \)
67 \( 1 + (0.239 + 0.970i)T^{2} \)
71 \( 1 + (-0.464 + 0.885i)T^{2} \)
73 \( 1 + (-0.935 - 0.354i)T^{2} \)
79 \( 1 + (-0.0359 + 0.593i)T + (-0.992 - 0.120i)T^{2} \)
83 \( 1 + (1.40 + 1.40i)T + iT^{2} \)
89 \( 1 + (-0.354 + 0.935i)T^{2} \)
97 \( 1 + (-0.568 + 0.822i)T^{2} \)
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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

−9.095244422202913345855899370981, −8.252911723942391369079267859271, −7.66313588599315745951366431442, −6.82923730374100024852736173667, −6.02861485899934860772545707563, −5.33069723766634228845611754349, −4.66250490460406919758017236712, −3.40914457385602578794060368506, −2.29312658549316063587468870372, −1.27640884005111521869596845518, 1.46442545647634123029660167782, 2.53593080436798010126784061043, 3.13997899240565897296388195960, 4.06424811876752271780018862974, 5.30840366395604191106005674429, 6.17603124532711468109967746832, 6.97105880657911668277449235371, 7.29886434450623946883850810037, 8.364414297741997352984081898520, 9.403129043148983252772540334972

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