Properties

Label 2-1859-143.120-c0-0-6
Degree $2$
Conductor $1859$
Sign $0.668 - 0.743i$
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 + 1.80·5-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)11-s − 0.445·12-s + (0.400 + 0.694i)15-s + (−0.499 − 0.866i)16-s + (−0.900 + 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + (−0.222 + 0.385i)33-s + (0.400 + 0.694i)36-s + (−0.222 − 0.385i)37-s + ⋯
L(s)  = 1  + (0.222 + 0.385i)3-s + (−0.5 + 0.866i)4-s + 1.80·5-s + (0.400 − 0.694i)9-s + (0.5 + 0.866i)11-s − 0.445·12-s + (0.400 + 0.694i)15-s + (−0.499 − 0.866i)16-s + (−0.900 + 1.56i)20-s + (−0.623 − 1.07i)23-s + 2.24·25-s + 0.801·27-s − 1.24·31-s + (−0.222 + 0.385i)33-s + (0.400 + 0.694i)36-s + (−0.222 − 0.385i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.668 - 0.743i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1836, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ 0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.555091241\)
\(L(\frac12)\) \(\approx\) \(1.555091241\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.557533532412402410412094291256, −8.999513583682295134449595311057, −8.182376216818277910501064590618, −6.95979408917485322469852805493, −6.50643633052932050019644965245, −5.41081938756247366711609829591, −4.57585311094722520186872946008, −3.74213256637403067702302565758, −2.66320411705426773043759822256, −1.65550912931061793235162238534, 1.47573279448361349308033880185, 1.90621716319244380771349774574, 3.27803989427556190133315264290, 4.70851902956968859427785684825, 5.40685512322721334723051482898, 6.09079546454994610816447797243, 6.65514215230311615052708103761, 7.83822822853876985496018895545, 8.755102995600120555998996211888, 9.505665489637455148315344149447

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