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Properties

Label 2-1872-13.9-c1-0-18
Degree $2$
Conductor $1872$
Sign $0.923 - 0.384i$
Analytic cond. $14.9479$
Root an. cond. $3.86626$
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  + 3.56·5-s + (0.280 + 0.486i)7-s + (1 − 1.73i)11-s + (0.5 + 3.57i)13-s + (−0.780 − 1.35i)17-s + (3.56 + 6.16i)19-s + (−1 + 1.73i)23-s + 7.68·25-s + (3.34 − 5.78i)29-s − 2.56·31-s + (1 + 1.73i)35-s + (−3.78 + 6.54i)37-s + (−0.780 + 1.35i)41-s + (2.28 + 3.95i)43-s + 8.24·47-s + ⋯
L(s)  = 1  + 1.59·5-s + (0.106 + 0.183i)7-s + (0.301 − 0.522i)11-s + (0.138 + 0.990i)13-s + (−0.189 − 0.327i)17-s + (0.817 + 1.41i)19-s + (−0.208 + 0.361i)23-s + 1.53·25-s + (0.620 − 1.07i)29-s − 0.460·31-s + (0.169 + 0.292i)35-s + (−0.621 + 1.07i)37-s + (−0.121 + 0.211i)41-s + (0.347 + 0.602i)43-s + 1.20·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1872\)    =    \(2^{4} \cdot 3^{2} \cdot 13\)
Sign: $0.923 - 0.384i$
Analytic conductor: \(14.9479\)
Root analytic conductor: \(3.86626\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1872} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1872,\ (\ :1/2),\ 0.923 - 0.384i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.497635249\)
\(L(\frac12)\) \(\approx\) \(2.497635249\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + (-0.5 - 3.57i)T \)
good5 \( 1 - 3.56T + 5T^{2} \)
7 \( 1 + (-0.280 - 0.486i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.780 + 1.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.56 - 6.16i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.34 + 5.78i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.56T + 31T^{2} \)
37 \( 1 + (3.78 - 6.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.780 - 1.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.28 - 3.95i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 - 0.684T + 53T^{2} \)
59 \( 1 + (-1.43 - 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.93 + 3.35i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.28 + 3.95i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 + 5.43T + 79T^{2} \)
83 \( 1 + 0.876T + 83T^{2} \)
89 \( 1 + (-2.43 + 4.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.28 - 7.41i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.325103184043324238972951115550, −8.720609449557943700477363563860, −7.73043287595750612361067722461, −6.68840465760489197477717599095, −6.01570562172815349605657951438, −5.48304244609786335471487191377, −4.42263564723302872281811982381, −3.27499667462837918667664030543, −2.14575147941085599467771918985, −1.34604065011123697457829307571, 1.04632549379668873469767006555, 2.17561951429215874973104856902, 3.03440136362793818258010634424, 4.35725988164092933301003533264, 5.35807984818000094637297283279, 5.80190402727421685797123992296, 6.86740871666235156099354333530, 7.38289184339981563565167438414, 8.747803213252725408541094553106, 9.087803291007023319471308564400

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