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Properties

Label 2-2376-9.4-c1-0-11
Degree $2$
Conductor $2376$
Sign $0.766 - 0.642i$
Analytic cond. $18.9724$
Root an. cond. $4.35573$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)11-s + (−1 + 1.73i)13-s − 6·17-s + 6·19-s + (−3.5 + 6.06i)23-s + (2 + 3.46i)25-s + (5 + 8.66i)29-s + 2·37-s + (6 − 10.3i)41-s + (4 + 6.92i)43-s + (0.5 + 0.866i)47-s + (3.5 − 6.06i)49-s + 9·53-s − 0.999·55-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (−0.150 − 0.261i)11-s + (−0.277 + 0.480i)13-s − 1.45·17-s + 1.37·19-s + (−0.729 + 1.26i)23-s + (0.400 + 0.692i)25-s + (0.928 + 1.60i)29-s + 0.328·37-s + (0.937 − 1.62i)41-s + (0.609 + 1.05i)43-s + (0.0729 + 0.126i)47-s + (0.5 − 0.866i)49-s + 1.23·53-s − 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2376\)    =    \(2^{3} \cdot 3^{3} \cdot 11\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(18.9724\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2376} (793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2376,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.641736988\)
\(L(\frac12)\) \(\approx\) \(1.641736988\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-6 + 10.3i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 9T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (-7 - 12.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 13T + 89T^{2} \)
97 \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.137453281004785815587701126634, −8.424110225154385623216658514168, −7.38806670701974831035492701545, −6.90396864195286646832731457995, −5.78840495681244345299561421937, −5.18641532741203251628010489410, −4.29007653973748002774471737284, −3.31278214715635285807320427463, −2.23120788569265401248427978338, −1.09514316245703514005357124702, 0.63824393014040962981170178140, 2.29388214693793777646647371580, 2.82916760170696193335109462407, 4.20611728668552518560263234701, 4.77223908191217599324203698637, 5.96101061516007858782795754759, 6.45103959340820889210844749968, 7.41042953617799825676449204717, 8.045115157996152533826734686322, 8.923665497516118641297585453029

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