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Properties

Label 2-2376-9.4-c1-0-23
Degree $2$
Conductor $2376$
Sign $-0.342 + 0.939i$
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  + (−0.866 + 1.5i)5-s + (−1.36 − 2.36i)7-s + (0.5 + 0.866i)11-s + (−0.633 + 1.09i)13-s + 1.26·17-s − 2·19-s + (−0.267 + 0.464i)23-s + (1 + 1.73i)25-s + (−1.36 − 2.36i)29-s + (2.23 − 3.86i)31-s + 4.73·35-s + 1.19·37-s + (0.366 − 0.633i)41-s + (−5.36 − 9.29i)43-s + (−5.69 − 9.86i)47-s + ⋯
L(s)  = 1  + (−0.387 + 0.670i)5-s + (−0.516 − 0.894i)7-s + (0.150 + 0.261i)11-s + (−0.175 + 0.304i)13-s + 0.307·17-s − 0.458·19-s + (−0.0558 + 0.0967i)23-s + (0.200 + 0.346i)25-s + (−0.253 − 0.439i)29-s + (0.400 − 0.694i)31-s + 0.799·35-s + 0.196·37-s + (0.0571 − 0.0990i)41-s + (−0.818 − 1.41i)43-s + (−0.830 − 1.43i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2376\)    =    \(2^{3} \cdot 3^{3} \cdot 11\)
Sign: $-0.342 + 0.939i$
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: \((2,\ 2376,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7292922831\)
\(L(\frac12)\) \(\approx\) \(0.7292922831\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 1.5i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.36 + 2.36i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (0.633 - 1.09i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.26T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + (0.267 - 0.464i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.36 + 2.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.23 + 3.86i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.19T + 37T^{2} \)
41 \( 1 + (-0.366 + 0.633i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.36 + 9.29i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.69 + 9.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6.66T + 53T^{2} \)
59 \( 1 + (6.59 - 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.535 - 0.928i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.59 + 6.23i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.46T + 71T^{2} \)
73 \( 1 - 0.196T + 73T^{2} \)
79 \( 1 + (6.83 + 11.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2 + 3.46i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 2.53T + 89T^{2} \)
97 \( 1 + (5.96 + 10.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.718498673265444306228170996522, −7.81212011762435461669127605801, −7.11341155247323538745964128433, −6.65904309036322067904722668529, −5.69596704781927095415095337203, −4.56116709691377280334789852502, −3.80345889051216491341521291711, −3.06831866744077650551603393261, −1.83126813394956781970555670277, −0.26057856178796175518300626233, 1.23141150985696966150117374245, 2.60473679058207473710558507818, 3.41006746392434507180024550133, 4.53472480155370177206799699678, 5.20746229789524530419180760548, 6.14282676693483360525480457509, 6.73482084660094457856397217542, 8.025888212087428344353103944512, 8.292547708260172039423541398133, 9.263539590647690983071398549567

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