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Properties

Label 2-1859-143.43-c0-0-3
Degree $2$
Conductor $1859$
Sign $-0.0841 - 0.996i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.0841 - 0.996i$
Analytic conductor: \(0.927761\)
Root analytic conductor: \(0.963203\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (1330, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :0),\ -0.0841 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321358686\)
\(L(\frac12)\) \(\approx\) \(1.321358686\)
\(L(1)\) 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

−9.836121540447702744102244634156, −8.523027891310658657483399328980, −7.80706665807216411458056824282, −7.38574439420695273760467249850, −6.64880808964868482592973623947, −5.99020794847241630383264158244, −4.59067060042079895066169879119, −3.45111173123523100998295900913, −2.70737644920416850123003698052, −2.19162991170558254792496899730, 0.973690227844923644852806549186, 1.92979755451323744700757625870, 3.37868366659555026625981492731, 4.59521520427066040938114567915, 5.05260563768024283280173942444, 5.78814758541660149144394510610, 6.84210154953640562488753034833, 7.73150496147890810044093591372, 8.628441664270635884380101183807, 9.413570665484435660059403445735

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