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Properties

Label 2-3328-104.5-c0-0-0
Degree $2$
Conductor $3328$
Sign $0.471 - 0.881i$
Analytic cond. $1.66088$
Root an. cond. $1.28875$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + i)5-s + 9-s − 13-s + 2i·17-s + i·25-s + (−1 + i)37-s + (1 − i)41-s + (1 + i)45-s − i·49-s − 2i·53-s + 2i·61-s + (−1 − i)65-s + (−1 − i)73-s + 81-s + (−2 + 2i)85-s + ⋯
L(s)  = 1  + (1 + i)5-s + 9-s − 13-s + 2i·17-s + i·25-s + (−1 + i)37-s + (1 − i)41-s + (1 + i)45-s − i·49-s − 2i·53-s + 2i·61-s + (−1 − i)65-s + (−1 − i)73-s + 81-s + (−2 + 2i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(1.66088\)
Root analytic conductor: \(1.28875\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.542862390\)
\(L(\frac12)\) \(\approx\) \(1.542862390\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + T \)
good3 \( 1 - T^{2} \)
5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 - iT^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (1 - i)T - iT^{2} \)
41 \( 1 + (-1 + i)T - iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 - 2iT - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.020335316488949090333019129400, −8.146551650169916998430184850327, −7.25776425857025797586048360270, −6.72213442777224792575476016884, −6.04587564502021102708616966066, −5.24442142324189201626283391205, −4.23463749079877553814955314773, −3.39502944110439695414161423053, −2.27981476099698852736146140868, −1.65252424423980166848688918642, 0.964470199143662679794386037296, 2.02013754100662351469489891487, 2.90900387350512564520107225920, 4.31698359375137969189492398652, 4.89480070284721819824701950521, 5.46444559557286676739719281518, 6.43261251752249387085272776476, 7.30560468510762062220321870330, 7.76530354570587196059614796424, 9.084828647908634588857275872598

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