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Properties

Label 2-837-93.68-c1-0-31
Degree $2$
Conductor $837$
Sign $0.390 + 0.920i$
Analytic cond. $6.68347$
Root an. cond. $2.58524$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·4-s + (−2.5 − 4.33i)7-s + (4.5 + 2.59i)13-s + 4·16-s + (−3.5 − 6.06i)19-s + (−2.5 − 4.33i)25-s + (−5 − 8.66i)28-s + (2 − 5.19i)31-s + (4.5 − 2.59i)37-s + (1.5 − 0.866i)43-s + (−9.00 + 15.5i)49-s + (9 + 5.19i)52-s + 8.66i·61-s + 8·64-s + (5.5 − 9.52i)67-s + ⋯
L(s)  = 1  + 4-s + (−0.944 − 1.63i)7-s + (1.24 + 0.720i)13-s + 16-s + (−0.802 − 1.39i)19-s + (−0.5 − 0.866i)25-s + (−0.944 − 1.63i)28-s + (0.359 − 0.933i)31-s + (0.739 − 0.427i)37-s + (0.228 − 0.132i)43-s + (−1.28 + 2.22i)49-s + (1.24 + 0.720i)52-s + 1.10i·61-s + 64-s + (0.671 − 1.16i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.390 + 0.920i$
Analytic conductor: \(6.68347\)
Root analytic conductor: \(2.58524\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 837,\ (\ :1/2),\ 0.390 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45616 - 0.964131i\)
\(L(\frac12)\) \(\approx\) \(1.45616 - 0.964131i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 + (-2 + 5.19i)T \)
good2 \( 1 - 2T^{2} \)
5 \( 1 + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.5 + 4.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + (-4.5 + 2.59i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.5 + 0.866i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 + (-5.5 + 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-12 - 6.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (10.5 - 6.06i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 14T + 97T^{2} \)
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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

−10.19946170759739813632001890115, −9.366647916484757163315323806657, −8.212228305222830744093237394997, −7.26491832133683274984688223973, −6.60186605864454167863520068670, −6.08606564867199397062043696805, −4.35630763141934893195528245498, −3.65328169563820756027401966491, −2.41555362828993481173566442498, −0.861651706826499882621914749961, 1.71551518566456141501732307859, 2.86540413891020270540426299325, 3.62708941106990905197283893945, 5.48253682978708830472600486482, 6.03316770637505305075552413659, 6.63472160441853148145316048406, 7.966982487468075399162842259388, 8.571603791360317168259616792372, 9.573793682984379580503469082135, 10.38906827642735866738781604118

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