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Properties

Label 2-2128-532.75-c0-0-4
Degree $2$
Conductor $2128$
Sign $0.605 - 0.795i$
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 + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)11-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 + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + 3·55-s + (−1.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯
L(s)  = 1  + (1.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s + (−0.5 + 0.866i)9-s + (1.5 − 0.866i)11-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 + (−0.5 + 0.866i)47-s + (−0.499 − 0.866i)49-s + 3·55-s + (−1.5 − 0.866i)61-s + (−0.499 − 0.866i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2128\)    =    \(2^{4} \cdot 7 \cdot 19\)
Sign: $0.605 - 0.795i$
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: \((2,\ 2128,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.486679321\)
\(L(\frac12)\) \(\approx\) \(1.486679321\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
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 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
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

−9.421584615111340830512321215627, −8.830654638006002333635792837989, −7.932388157250182007583918749478, −6.68239033022775538223791710951, −6.19356564175547818110603539975, −5.77093257515846501185404354096, −4.71215673595669593516441236667, −3.27891043958527578360939212951, −2.61463658342847155891111444516, −1.72080273339896398541295768904, 1.18230460808785983394956349765, 1.98778001504534548398865043734, 3.55317679433870760296363359254, 4.16926382172128054001661105122, 5.33911680886528468558921014472, 6.08451828908086746233396723287, 6.60070047556043472933429782229, 7.52554980726921347994747031501, 8.708506787600451563904095291912, 9.262675324269625945857583643064

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