Properties

Degree $2$
Conductor $5376$
Sign $0.707 - 0.707i$
Motivic weight $1$
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 + 2i·13-s − 2·15-s − 6·17-s − 4i·19-s i·21-s + 25-s + i·27-s + 2i·29-s + 4·33-s − 2i·35-s + ⋯
L(s)  = 1  − 0.577i·3-s − 0.894i·5-s + 0.377·7-s − 0.333·9-s + 1.20i·11-s + 0.554i·13-s − 0.516·15-s − 1.45·17-s − 0.917i·19-s − 0.218i·21-s + 0.200·25-s + 0.192i·27-s + 0.371i·29-s + 0.696·33-s − 0.338i·35-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: $0.707 - 0.707i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(1.275947048\)
\(L(\frac12)\) \(\approx\) \(1.275947048\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 2iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 12iT - 59T^{2} \)
61 \( 1 - 2iT - 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 14T + 89T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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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

−8.350517693041277270735599658787, −7.44656064467612503377806834424, −6.91500810570979008521380896108, −6.29349622223517076572567579270, −5.12645213216649848307893433446, −4.72792771328861545304698602156, −4.05094925871767125867850116134, −2.64875407948134032120680640951, −1.93209494769392056603312768058, −1.04504733105666765977175647277, 0.36009807948245557570723673684, 1.91973623200180722729461837883, 2.88956951127148359134542731072, 3.52254486113530012337037724094, 4.32335758214917138159052045224, 5.21765163285796894274478632993, 5.98080186583108787327345899579, 6.53224710163648512416763596229, 7.40039972237262266761801591035, 8.229852418487187662755735810790

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