Properties

Label 2-2888-2888.2771-c0-0-0
Degree $2$
Conductor $2888$
Sign $0.994 + 0.104i$
Analytic cond. $1.44129$
Root an. cond. $1.20054$
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  + (−0.649 − 0.760i)2-s + (−1.53 − 1.14i)3-s + (−0.155 + 0.987i)4-s + (0.123 + 1.91i)6-s + (0.851 − 0.523i)8-s + (0.748 + 2.55i)9-s + (−1.98 − 0.219i)11-s + (1.37 − 1.33i)12-s + (−0.951 − 0.307i)16-s + (−1.72 − 0.665i)17-s + (1.45 − 2.23i)18-s + (−0.333 − 0.942i)19-s + (1.12 + 1.65i)22-s + (−1.90 − 0.175i)24-s + (0.100 − 0.994i)25-s + ⋯
L(s)  = 1  + (−0.649 − 0.760i)2-s + (−1.53 − 1.14i)3-s + (−0.155 + 0.987i)4-s + (0.123 + 1.91i)6-s + (0.851 − 0.523i)8-s + (0.748 + 2.55i)9-s + (−1.98 − 0.219i)11-s + (1.37 − 1.33i)12-s + (−0.951 − 0.307i)16-s + (−1.72 − 0.665i)17-s + (1.45 − 2.23i)18-s + (−0.333 − 0.942i)19-s + (1.12 + 1.65i)22-s + (−1.90 − 0.175i)24-s + (0.100 − 0.994i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2888\)    =    \(2^{3} \cdot 19^{2}\)
Sign: $0.994 + 0.104i$
Analytic conductor: \(1.44129\)
Root analytic conductor: \(1.20054\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2888} (2771, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.994 + 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1672292825\)
\(L(\frac12)\) \(\approx\) \(0.1672292825\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.649 + 0.760i)T \)
19 \( 1 + (0.333 + 0.942i)T \)
good3 \( 1 + (1.53 + 1.14i)T + (0.280 + 0.959i)T^{2} \)
5 \( 1 + (-0.100 + 0.994i)T^{2} \)
7 \( 1 + (-0.137 - 0.990i)T^{2} \)
11 \( 1 + (1.98 + 0.219i)T + (0.975 + 0.218i)T^{2} \)
13 \( 1 + (0.896 - 0.443i)T^{2} \)
17 \( 1 + (1.72 + 0.665i)T + (0.741 + 0.670i)T^{2} \)
23 \( 1 + (-0.690 - 0.723i)T^{2} \)
29 \( 1 + (0.467 + 0.883i)T^{2} \)
31 \( 1 + (0.191 - 0.981i)T^{2} \)
37 \( 1 + (0.677 - 0.735i)T^{2} \)
41 \( 1 + (-0.384 - 1.40i)T + (-0.861 + 0.507i)T^{2} \)
43 \( 1 + (-0.425 - 0.557i)T + (-0.263 + 0.964i)T^{2} \)
47 \( 1 + (-0.606 - 0.794i)T^{2} \)
53 \( 1 + (-0.384 + 0.922i)T^{2} \)
59 \( 1 + (1.22 - 1.23i)T + (-0.00918 - 0.999i)T^{2} \)
61 \( 1 + (0.227 - 0.973i)T^{2} \)
67 \( 1 + (-1.90 + 0.390i)T + (0.919 - 0.393i)T^{2} \)
71 \( 1 + (0.227 + 0.973i)T^{2} \)
73 \( 1 + (-0.409 + 0.330i)T + (0.209 - 0.977i)T^{2} \)
79 \( 1 + (0.703 + 0.710i)T^{2} \)
83 \( 1 + (-0.866 - 0.352i)T + (0.716 + 0.697i)T^{2} \)
89 \( 1 + (0.213 + 0.172i)T + (0.209 + 0.977i)T^{2} \)
97 \( 1 + (0.340 + 0.0697i)T + (0.919 + 0.393i)T^{2} \)
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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

−8.874900417298624968801308924966, −7.998823867462489157947179533619, −7.53560085560194294466114443318, −6.73859489505997464521451820294, −6.08255852143214567471416388535, −4.87988186450157332681314540416, −4.62626831131722651703711632160, −2.64358850586959877291722485709, −2.26837307151485833388843382624, −0.793883091884778895752051817969, 0.22500034303507223801663628072, 2.06034382047632025072767361692, 3.78319186936813741914957321187, 4.65603237958613901653726000534, 5.28599532765829489116931140663, 5.78673521470415738212910382668, 6.57846395318421064909076877180, 7.31252851362009679640179066106, 8.291802991077829513837346426765, 9.038584178989670764050121592101

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