Properties

Label 2-2394-21.20-c1-0-33
Degree $2$
Conductor $2394$
Sign $0.621 + 0.783i$
Analytic cond. $19.1161$
Root an. cond. $4.37220$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + 4.23·5-s + (2.64 − 0.145i)7-s + i·8-s − 4.23i·10-s − 0.875i·11-s + 3.49i·13-s + (−0.145 − 2.64i)14-s + 16-s + 0.180·17-s i·19-s − 4.23·20-s − 0.875·22-s + 1.13i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.89·5-s + (0.998 − 0.0549i)7-s + 0.353i·8-s − 1.33i·10-s − 0.264i·11-s + 0.970i·13-s + (−0.0388 − 0.706i)14-s + 0.250·16-s + 0.0437·17-s − 0.229i·19-s − 0.946·20-s − 0.186·22-s + 0.237i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2394\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $0.621 + 0.783i$
Analytic conductor: \(19.1161\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2394} (2015, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2394,\ (\ :1/2),\ 0.621 + 0.783i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.836190570\)
\(L(\frac12)\) \(\approx\) \(2.836190570\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 + (-2.64 + 0.145i)T \)
19 \( 1 + iT \)
good5 \( 1 - 4.23T + 5T^{2} \)
11 \( 1 + 0.875iT - 11T^{2} \)
13 \( 1 - 3.49iT - 13T^{2} \)
17 \( 1 - 0.180T + 17T^{2} \)
23 \( 1 - 1.13iT - 23T^{2} \)
29 \( 1 + 9.84iT - 29T^{2} \)
31 \( 1 + 1.21iT - 31T^{2} \)
37 \( 1 + 6.29T + 37T^{2} \)
41 \( 1 - 8.20T + 41T^{2} \)
43 \( 1 - 0.254T + 43T^{2} \)
47 \( 1 + 8.88T + 47T^{2} \)
53 \( 1 - 10.5iT - 53T^{2} \)
59 \( 1 - 13.0T + 59T^{2} \)
61 \( 1 + 6.24iT - 61T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 - 3.26iT - 71T^{2} \)
73 \( 1 - 16.2iT - 73T^{2} \)
79 \( 1 + 12.6T + 79T^{2} \)
83 \( 1 - 10.6T + 83T^{2} \)
89 \( 1 - 1.87T + 89T^{2} \)
97 \( 1 + 6.10iT - 97T^{2} \)
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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

−9.030637002040069153260342306320, −8.400768472187663526044883255930, −7.31634160967802930785193281671, −6.30783070348024428286024080625, −5.65703937256916245824372836623, −4.88929021433077389919796781910, −4.06747630652854312042727365732, −2.63397936810007466391134907221, −2.01239705274977261161326457109, −1.18527545536550587898992647838, 1.24236342253136298369036690742, 2.12549261058558133543199312083, 3.27619846883384820413340624261, 4.76994774865087744727876517764, 5.26558462794900114690071442501, 5.85161064285235995126250249400, 6.69572444699282980406609546590, 7.44418535351852112182137122358, 8.438609474796705216005544489920, 8.920471989069147563114056032297

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