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Properties

Label 2-1856-29.12-c0-0-1
Degree $2$
Conductor $1856$
Sign $0.189 + 0.981i$
Analytic cond. $0.926264$
Root an. cond. $0.962426$
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  + (−0.707 − 0.707i)3-s − i·5-s + 1.41·7-s + (0.707 + 0.707i)11-s − i·13-s + (−0.707 + 0.707i)15-s + (1.41 + 1.41i)19-s + (−1.00 − 1.00i)21-s + (−0.707 + 0.707i)27-s + 29-s + (−0.707 − 0.707i)31-s − 1.00i·33-s − 1.41i·35-s + (−0.707 + 0.707i)39-s + (−1 + i)41-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)3-s − i·5-s + 1.41·7-s + (0.707 + 0.707i)11-s − i·13-s + (−0.707 + 0.707i)15-s + (1.41 + 1.41i)19-s + (−1.00 − 1.00i)21-s + (−0.707 + 0.707i)27-s + 29-s + (−0.707 − 0.707i)31-s − 1.00i·33-s − 1.41i·35-s + (−0.707 + 0.707i)39-s + (−1 + i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $0.189 + 0.981i$
Analytic conductor: \(0.926264\)
Root analytic conductor: \(0.962426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :0),\ 0.189 + 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.129582446\)
\(L(\frac12)\) \(\approx\) \(1.129582446\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
5 \( 1 + iT - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
23 \( 1 + T^{2} \)
31 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1 - i)T - iT^{2} \)
43 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
47 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
53 \( 1 + T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (1 + i)T + iT^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−9.248753602154198421883263368960, −8.175360845627170479375954636877, −7.86498746619728676973211476203, −6.92184617420595263776580664265, −5.92756959032073291770971489426, −5.21531078064734730499977978158, −4.63494130675863486190679816116, −3.44270381947359127555374221822, −1.65665371201360579663008121404, −1.15712583145774471288424075999, 1.52535557095759612137652488576, 2.84750501483279464119216878026, 3.91687094722204743879520206247, 4.90194383161696158555873562598, 5.29469253483788620185800293119, 6.54237116977514472278976771192, 7.04228802911471374114553152831, 8.060817189816811198252882864515, 8.884845305887680702132560683424, 9.689727669945959160292605736323

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