Properties

Label 2-2888-2888.2707-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  + (−0.00918 − 0.999i)2-s + (1.75 + 0.532i)3-s + (−0.999 + 0.0183i)4-s + (0.516 − 1.76i)6-s + (0.0275 + 0.999i)8-s + (1.98 + 1.32i)9-s + (−1.10 + 1.50i)11-s + (−1.76 − 0.500i)12-s + (0.999 − 0.0367i)16-s + (0.132 + 0.240i)17-s + (1.30 − 1.99i)18-s + (0.870 − 0.492i)19-s + (1.51 + 1.09i)22-s + (−0.484 + 1.77i)24-s + (−0.435 + 0.900i)25-s + ⋯
L(s)  = 1  + (−0.00918 − 0.999i)2-s + (1.75 + 0.532i)3-s + (−0.999 + 0.0183i)4-s + (0.516 − 1.76i)6-s + (0.0275 + 0.999i)8-s + (1.98 + 1.32i)9-s + (−1.10 + 1.50i)11-s + (−1.76 − 0.500i)12-s + (0.999 − 0.0367i)16-s + (0.132 + 0.240i)17-s + (1.30 − 1.99i)18-s + (0.870 − 0.492i)19-s + (1.51 + 1.09i)22-s + (−0.484 + 1.77i)24-s + (−0.435 + 0.900i)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: $\chi_{2888} (2707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2888,\ (\ :0),\ 0.994 + 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.963188425\)
\(L(\frac12)\) \(\approx\) \(1.963188425\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.00918 + 0.999i)T \)
19 \( 1 + (-0.870 + 0.492i)T \)
good3 \( 1 + (-1.75 - 0.532i)T + (0.832 + 0.554i)T^{2} \)
5 \( 1 + (0.435 - 0.900i)T^{2} \)
7 \( 1 + (0.926 + 0.376i)T^{2} \)
11 \( 1 + (1.10 - 1.50i)T + (-0.298 - 0.954i)T^{2} \)
13 \( 1 + (0.971 + 0.236i)T^{2} \)
17 \( 1 + (-0.132 - 0.240i)T + (-0.531 + 0.847i)T^{2} \)
23 \( 1 + (0.896 + 0.443i)T^{2} \)
29 \( 1 + (0.951 - 0.307i)T^{2} \)
31 \( 1 + (0.754 + 0.656i)T^{2} \)
37 \( 1 + (0.677 - 0.735i)T^{2} \)
41 \( 1 + (0.924 + 1.30i)T + (-0.333 + 0.942i)T^{2} \)
43 \( 1 + (-1.12 + 0.584i)T + (0.577 - 0.816i)T^{2} \)
47 \( 1 + (-0.888 + 0.459i)T^{2} \)
53 \( 1 + (0.0459 - 0.998i)T^{2} \)
59 \( 1 + (-1.29 + 0.119i)T + (0.983 - 0.182i)T^{2} \)
61 \( 1 + (0.119 + 0.992i)T^{2} \)
67 \( 1 + (0.0798 - 0.100i)T + (-0.227 - 0.973i)T^{2} \)
71 \( 1 + (0.119 - 0.992i)T^{2} \)
73 \( 1 + (-0.595 + 0.989i)T + (-0.467 - 0.883i)T^{2} \)
79 \( 1 + (0.995 + 0.0917i)T^{2} \)
83 \( 1 + (0.261 + 1.88i)T + (-0.962 + 0.272i)T^{2} \)
89 \( 1 + (0.955 + 1.58i)T + (-0.467 + 0.883i)T^{2} \)
97 \( 1 + (-1.16 - 1.47i)T + (-0.227 + 0.973i)T^{2} \)
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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

−9.141142830846376145380818842111, −8.403593959982768472356762145675, −7.63606267047770318952267033594, −7.21937898524047691624950958722, −5.35471818955013677441808806451, −4.76549809689472680361907148155, −3.90989786660171834908566848589, −3.21034394847814771717102601567, −2.37215099463410916207310473635, −1.76263991542528062447062777695, 1.10788722447464078896026536603, 2.64036115970928423336041799177, 3.28322850883167015991250600819, 4.10107737141079392898060909920, 5.24198015893015617040840131651, 6.08105018530287749064807405901, 6.92447034448421722167390867466, 7.71899584076553226225628532179, 8.267771852384874332084712725118, 8.438919973047126092269798171699

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