Properties

Label 2-5376-8.5-c1-0-23
Degree $2$
Conductor $5376$
Sign $-0.707 - 0.707i$
Analytic cond. $42.9275$
Root an. cond. $6.55191$
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·3-s + 2i·5-s + 7-s − 9-s + 4i·11-s − 6i·13-s − 2·15-s − 2·17-s + 4i·19-s + i·21-s + 4·23-s + 25-s i·27-s − 2i·29-s + 8·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.894i·5-s + 0.377·7-s − 0.333·9-s + 1.20i·11-s − 1.66i·13-s − 0.516·15-s − 0.485·17-s + 0.917i·19-s + 0.218i·21-s + 0.834·23-s + 0.200·25-s − 0.192i·27-s − 0.371i·29-s + 1.43·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5376\)    =    \(2^{8} \cdot 3 \cdot 7\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(42.9275\)
Root analytic conductor: \(6.55191\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{5376} (2689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5376,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.744581748\)
\(L(\frac12)\) \(\approx\) \(1.744581748\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 - T \)
good5 \( 1 - 2iT - 5T^{2} \)
11 \( 1 - 4iT - 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 10iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 10iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.364146612788316031959783260767, −7.72408032474548797665328768976, −7.08490668393334585825389299390, −6.28592909289214005307878865614, −5.50892827791143925804181556418, −4.75110716284529228567779192130, −4.07258768172101631531743012042, −2.99871168567203333399958356231, −2.58048010489953929843379145581, −1.20526511451881696966041133531, 0.50083758549259891800268889200, 1.39131758895307238553349666187, 2.33602292609139606580468359735, 3.32321065157573113619754379714, 4.46061011725868076362322249460, 4.85461910560823371059418188057, 5.80683555839617801561362708889, 6.57003800194958448515494257296, 7.09105923397359768950600885078, 8.039410678429660950698247707711

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