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Properties

Label 2-2128-532.75-c0-0-5
Degree $2$
Conductor $2128$
Sign $0.386 + 0.922i$
Analytic cond. $1.06201$
Root an. cond. $1.03053$
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  + (−1.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1.5 − 0.866i)23-s + (1 + 1.73i)25-s + (−1.5 − 0.866i)35-s − 1.73i·43-s + (1.5 − 0.866i)45-s + (1 − 1.73i)47-s + 49-s + (−0.5 + 0.866i)63-s + (−0.499 − 0.866i)81-s − 83-s + ⋯
L(s)  = 1  + (−1.5 − 0.866i)5-s + 7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−1.5 − 0.866i)23-s + (1 + 1.73i)25-s + (−1.5 − 0.866i)35-s − 1.73i·43-s + (1.5 − 0.866i)45-s + (1 − 1.73i)47-s + 49-s + (−0.5 + 0.866i)63-s + (−0.499 − 0.866i)81-s − 83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2128\)    =    \(2^{4} \cdot 7 \cdot 19\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2128} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2128,\ (\ :0),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9013283135\)
\(L(\frac12)\) \(\approx\) \(0.9013283135\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73iT - T^{2} \)
47 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.659330615228117947383402275993, −8.470507690608842951038007020025, −7.57033008933390230494106598631, −7.30165757746488259783406120048, −5.62415852236648704299354914853, −5.05820642195303694774687866612, −4.37513906216833030932583372844, −3.46851343277349167427868613284, −2.21075619522109966132077998261, −0.72206824961062071798645446657, 1.35089012953080285445316301409, 2.95540018165320772393406118273, 3.69695136520859988420538150719, 4.27568027461664598340555195316, 5.61091398054100387808665124673, 6.22611235115667258567388439865, 7.40471070474306594963870286449, 7.911992987174439869646672830492, 8.246857951311914064867923837188, 9.447518534286920168368871786696

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