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Properties

Label 2-1859-143.10-c0-0-4
Degree $2$
Conductor $1859$
Sign $0.890 - 0.455i$
Analytic cond. $0.927761$
Root an. cond. $0.963203$
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  + (0.222 + 0.385i)3-s + (0.5 − 0.866i)4-s + 1.80i·5-s + (0.400 − 0.694i)9-s + (0.866 − 0.5i)11-s + 0.445·12-s + (−0.694 + 0.400i)15-s + (−0.499 − 0.866i)16-s + (1.56 + 0.900i)20-s + (0.623 + 1.07i)23-s − 2.24·25-s + 0.801·27-s − 1.24i·31-s + (0.385 + 0.222i)33-s + (−0.400 − 0.694i)36-s + (−0.385 + 0.222i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.385i)3-s + (0.5 − 0.866i)4-s + 1.80i·5-s + (0.400 − 0.694i)9-s + (0.866 − 0.5i)11-s + 0.445·12-s + (−0.694 + 0.400i)15-s + (−0.499 − 0.866i)16-s + (1.56 + 0.900i)20-s + (0.623 + 1.07i)23-s − 2.24·25-s + 0.801·27-s − 1.24i·31-s + (0.385 + 0.222i)33-s + (−0.400 − 0.694i)36-s + (−0.385 + 0.222i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.890 - 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.487467019\)
\(L(\frac12)\) \(\approx\) \(1.487467019\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - 1.80iT - T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.24iT - T^{2} \)
37 \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - 1.24iT - T^{2} \)
53 \( 1 + 1.80T + T^{2} \)
59 \( 1 + (-0.385 - 0.222i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.07 + 0.623i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.56 + 0.900i)T + (0.5 + 0.866i)T^{2} \)
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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

−9.647454127914674057343513166762, −9.033897380574399630040768822181, −7.59602050020837391824860133987, −7.04040991524606901398572057006, −6.25476982048119851396957850650, −5.87967771373879098467064669543, −4.41523618369482254243285754342, −3.42423741575937864993172411828, −2.79279637228783034137670805148, −1.48501976870867807202008471957, 1.36225303869284706303522324955, 2.18422802840412724414080244246, 3.57252081389335338897341488203, 4.55128081342638344216639750482, 5.03837793663575528986532904317, 6.36925914887158471323021776219, 7.15449695572697720850167118528, 7.88064721429151596640714846023, 8.684974015703302138042045219077, 8.923245345334708986724574512551

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