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Properties

Label 2-2128-133.18-c0-0-4
Degree $2$
Conductor $2128$
Sign $-0.895 + 0.444i$
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 − 1.73i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−1.49 + 2.59i)25-s − 1.99·35-s − 2·43-s + (−0.999 + 1.73i)45-s + (−0.5 − 0.866i)47-s + (−0.499 − 0.866i)49-s + 1.99·55-s + (0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−1 − 1.73i)5-s + (0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)23-s + (−1.49 + 2.59i)25-s − 1.99·35-s − 2·43-s + (−0.999 + 1.73i)45-s + (−0.5 − 0.866i)47-s + (−0.499 − 0.866i)49-s + 1.99·55-s + (0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2128\)    =    \(2^{4} \cdot 7 \cdot 19\)
Sign: $-0.895 + 0.444i$
Analytic conductor: \(1.06201\)
Root analytic conductor: \(1.03053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2128} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2128,\ (\ :0),\ -0.895 + 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7213694160\)
\(L(\frac12)\) \(\approx\) \(0.7213694160\)
\(L(1)\) 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 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 2T + T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.813419084123669020957093842692, −8.117961030384810626376934711731, −7.66422529883835850432066947395, −6.77677885578494512395463355894, −5.44550877662521242518424498477, −4.88610224268032303691317605872, −4.13482737897399672125272523849, −3.37400144630158216670578194209, −1.62723535878004301495587793038, −0.50808988783476093579504957568, 2.12231506943909371368472235352, 2.99957676295888914589009525769, 3.56711007551147313958582428282, 4.88660129707229311740319485909, 5.73740428519226720156244440612, 6.47438792942762629308543098369, 7.46080611846657440068459729531, 8.065844195349755557467447390175, 8.453448668459997295752894324619, 9.746940635088989142203619801509

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