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Properties

Label 2-1848-1848.1451-c0-0-6
Degree $2$
Conductor $1848$
Sign $-0.444 + 0.895i$
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  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s + 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.866 − 1.5i)5-s + 0.999·6-s + (−0.866 − 0.5i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.866 − 1.5i)10-s + (−0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.866 + 0.499i)14-s + 1.73·15-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (0.499 + 0.866i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.444 + 0.895i$
Analytic conductor: \(0.922272\)
Root analytic conductor: \(0.960350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (1451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :0),\ -0.444 + 0.895i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.579015054\)
\(L(\frac12)\) \(\approx\) \(1.579015054\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.365726318201083035204532799281, −8.712309785981611836270164721355, −8.106396720002475362052942591738, −6.34415283720117332580091293177, −5.66468528632071186415042876314, −4.91021681320875771222261709778, −4.20473267193311608976039701796, −3.28427308043161759260229441924, −2.36531100860884395627375750114, −0.954544830919603701012845111387, 2.18769429067931693795284223674, 2.94819059192874965927485293625, 3.52306069211554486161030569934, 5.23007625809876241700841557354, 5.90294768203319991174972429883, 6.64277278390667425719398360094, 7.23054557657680227019309722683, 7.61640946023312166364026781759, 8.893349173836755884257979572334, 9.554167104809092943359299141678

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