Properties

Label 2-1368-57.56-c1-0-8
Degree $2$
Conductor $1368$
Sign $0.927 - 0.374i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  − 1.41i·5-s + 2·7-s + 4.24i·11-s − 2.82i·13-s + 7.07i·17-s + (−1 + 4.24i)19-s + 1.41i·23-s + 2.99·25-s + 10·29-s − 2.82i·31-s − 2.82i·35-s − 5.65i·37-s − 10·41-s + 12·43-s − 1.41i·47-s + ⋯
L(s)  = 1  − 0.632i·5-s + 0.755·7-s + 1.27i·11-s − 0.784i·13-s + 1.71i·17-s + (−0.229 + 0.973i)19-s + 0.294i·23-s + 0.599·25-s + 1.85·29-s − 0.508i·31-s − 0.478i·35-s − 0.929i·37-s − 1.56·41-s + 1.82·43-s − 0.206i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.811134916\)
\(L(\frac12)\) \(\approx\) \(1.811134916\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (1 - 4.24i)T \)
good5 \( 1 + 1.41iT - 5T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 - 7.07iT - 17T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 - 10T + 29T^{2} \)
31 \( 1 + 2.82iT - 31T^{2} \)
37 \( 1 + 5.65iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + 1.41iT - 47T^{2} \)
53 \( 1 + 10T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 14.1iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 11.3iT - 79T^{2} \)
83 \( 1 - 4.24iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8.48iT - 97T^{2} \)
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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

−9.794016899796135216771393209884, −8.529328387751128566742027973583, −8.245422034110757923736611122994, −7.33774017461619826058639057211, −6.28618836418052364133872891072, −5.36092677339483047373607846600, −4.58365400720302668090992328073, −3.76360301192382975194235527835, −2.23487811188569162255459932050, −1.24744177128096154219579840111, 0.877217553245966555519678683096, 2.49920238489264669785081430596, 3.23133839037633065696402891294, 4.61228716912658158000555029225, 5.18084179701541969211734125613, 6.56367515045245335973639234107, 6.85067603869651007030626842406, 8.042002093302562572123530272945, 8.681729725480174863270102956246, 9.467833675955558206485723992858

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