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 − 3.46i·5-s + 7-s − 9-s + 5.46i·11-s − 2i·13-s − 3.46·15-s + 7.46·17-s + 6.92i·19-s i·21-s − 5.46·23-s − 6.99·25-s + i·27-s − 8.92i·29-s + 2.92·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.54i·5-s + 0.377·7-s − 0.333·9-s + 1.64i·11-s − 0.554i·13-s − 0.894·15-s + 1.81·17-s + 1.58i·19-s − 0.218i·21-s − 1.13·23-s − 1.39·25-s + 0.192i·27-s − 1.65i·29-s + 0.525·31-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: $-0.707 + 0.707i$
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

\(L(1)\) \(\approx\) \(1.709235969\)
\(L(\frac12)\) \(\approx\) \(1.709235969\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.46iT - 5T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 7.46T + 17T^{2} \)
19 \( 1 - 6.92iT - 19T^{2} \)
23 \( 1 + 5.46T + 23T^{2} \)
29 \( 1 + 8.92iT - 29T^{2} \)
31 \( 1 - 2.92T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 4.53T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 2.92T + 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 + 14.9iT - 59T^{2} \)
61 \( 1 + 4.92iT - 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 2.53T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 - 2.92T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 - 3.46T + 89T^{2} \)
97 \( 1 - 4.92T + 97T^{2} \)
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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

−8.049529663015836596798181728441, −7.54121793407339945642660611308, −6.38497683216916893888872649802, −5.59435029451244762726711532548, −5.12103723144253047213808126394, −4.29005085879892040976498223803, −3.53711592703475040868945207512, −2.01710352525635126858220231923, −1.59583053493359406601502906509, −0.48098954920790350812239599206, 1.16070893419317198944099399089, 2.65102661408536581996405329029, 3.14051916258183323447918206498, 3.76446421702001566996899511801, 4.82090247136823781326549190145, 5.67444671677751706845012387358, 6.24604238738743034252595965741, 7.00156849915268103893531365993, 7.69772673693921853967399350830, 8.464300163057408642349615321158

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