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Properties

Label 2-2352-147.8-c0-0-0
Degree $2$
Conductor $2352$
Sign $-0.0960 + 0.995i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (0.900 − 0.433i)3-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−1.12 − 1.40i)13-s − 1.24·19-s + (−0.900 − 0.433i)21-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)27-s + 0.445·31-s + (0.0990 + 0.433i)37-s + (−1.62 − 0.781i)39-s + (1.12 + 0.541i)43-s + (−0.222 + 0.974i)49-s + (−1.12 + 0.541i)57-s + (0.0990 + 0.433i)61-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)3-s + (−0.623 − 0.781i)7-s + (0.623 − 0.781i)9-s + (−1.12 − 1.40i)13-s − 1.24·19-s + (−0.900 − 0.433i)21-s + (0.623 − 0.781i)25-s + (0.222 − 0.974i)27-s + 0.445·31-s + (0.0990 + 0.433i)37-s + (−1.62 − 0.781i)39-s + (1.12 + 0.541i)43-s + (−0.222 + 0.974i)49-s + (−1.12 + 0.541i)57-s + (0.0990 + 0.433i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0960 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.0960 + 0.995i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.0960 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.336629315\)
\(L(\frac12)\) \(\approx\) \(1.336629315\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
good5 \( 1 + (-0.623 + 0.781i)T^{2} \)
11 \( 1 + (0.222 - 0.974i)T^{2} \)
13 \( 1 + (1.12 + 1.40i)T + (-0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.900 - 0.433i)T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + (0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 - 0.445T + T^{2} \)
37 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 + 0.433i)T^{2} \)
59 \( 1 + (-0.623 - 0.781i)T^{2} \)
61 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
67 \( 1 - 1.80T + T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
79 \( 1 - 0.445T + T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 + 0.974i)T^{2} \)
97 \( 1 - 1.24T + T^{2} \)
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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

−8.856827525413942035914480066492, −8.115065530399153127611294337293, −7.52743669092992618783938027979, −6.78078182853246940901449961460, −6.08658792092630831448680694525, −4.82261335313320895449011335087, −3.99433793550232347735033920213, −3.02044085189580903859284674834, −2.36041933230939673919589846582, −0.78758100573783104460665931067, 2.01446713119259575302048543417, 2.58282615850333624198874390332, 3.67551537772646229785362004713, 4.48900689667911986402465085393, 5.27875257700473989738706980221, 6.43941634275253273287329273889, 7.06427072135972070420013196652, 7.956164441738950363836071160618, 8.911842108755767480071126770602, 9.178970785600952678809396900105

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