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Properties

Label 2-1848-1848.317-c0-0-3
Degree $2$
Conductor $1848$
Sign $0.746 + 0.665i$
Analytic cond. $0.922272$
Root an. cond. $0.960350$
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  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (1.78 + 0.379i)5-s + (0.309 + 0.951i)6-s + (0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.913 − 1.58i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)14-s + (−1.47 − 1.07i)15-s + (−0.978 − 0.207i)16-s + (0.104 − 0.994i)18-s + ⋯
L(s)  = 1  + (−0.669 − 0.743i)2-s + (−0.913 − 0.406i)3-s + (−0.104 + 0.994i)4-s + (1.78 + 0.379i)5-s + (0.309 + 0.951i)6-s + (0.913 + 0.406i)7-s + (0.809 − 0.587i)8-s + (0.669 + 0.743i)9-s + (−0.913 − 1.58i)10-s + (0.5 − 0.866i)11-s + (0.5 − 0.866i)12-s + (−0.309 − 0.951i)14-s + (−1.47 − 1.07i)15-s + (−0.978 − 0.207i)16-s + (0.104 − 0.994i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.746 + 0.665i$
Analytic conductor: \(0.922272\)
Root analytic conductor: \(0.960350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :0),\ 0.746 + 0.665i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9614063721\)
\(L(\frac12)\) \(\approx\) \(0.9614063721\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.669 + 0.743i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (-1.78 - 0.379i)T + (0.913 + 0.406i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.104 + 0.994i)T^{2} \)
19 \( 1 + (0.978 - 0.207i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.169 + 0.122i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.30 - 0.278i)T + (0.913 - 0.406i)T^{2} \)
37 \( 1 + (-0.669 + 0.743i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.978 - 0.207i)T^{2} \)
53 \( 1 + (1.91 - 0.406i)T + (0.913 - 0.406i)T^{2} \)
59 \( 1 + (0.204 - 1.94i)T + (-0.978 - 0.207i)T^{2} \)
61 \( 1 + (-0.913 - 0.406i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.169 + 1.60i)T + (-0.978 - 0.207i)T^{2} \)
79 \( 1 + (0.139 + 0.155i)T + (-0.104 + 0.994i)T^{2} \)
83 \( 1 + (0.564 + 1.73i)T + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.0646 - 0.198i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.364405259279754464621792723408, −8.864625981048815928955888602646, −7.82800550940455285936151741893, −6.95368584948483292725228454971, −6.07968398365997428572245899737, −5.50525391234133290945800518617, −4.52208896638433188482545259654, −3.02516694049903998895767999263, −1.94186841794118877126554623514, −1.37489633938773338017489584793, 1.30866108692172255072864215127, 1.97233789949212835355168606944, 4.20292926105698614406527588833, 5.11337911117161987559739346274, 5.42332567761205505044448901628, 6.41327580480198995854098270929, 6.90571944893670489041520655876, 7.936861520827429531002662171666, 9.026087802187952772141805529504, 9.555648963170448628115773040844

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