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Properties

Label 2-4012-17.16-c1-0-80
Degree $2$
Conductor $4012$
Sign $-0.679 - 0.733i$
Analytic cond. $32.0359$
Root an. cond. $5.66003$
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  − 2.32i·3-s − 4.13i·5-s − 2.86i·7-s − 2.42·9-s + 0.105i·11-s + 4.79·13-s − 9.63·15-s + (−2.80 − 3.02i)17-s + 6.25·19-s − 6.66·21-s − 3.72i·23-s − 12.1·25-s − 1.34i·27-s − 0.0883i·29-s + 5.14i·31-s + ⋯
L(s)  = 1  − 1.34i·3-s − 1.84i·5-s − 1.08i·7-s − 0.807·9-s + 0.0318i·11-s + 1.32·13-s − 2.48·15-s + (−0.679 − 0.733i)17-s + 1.43·19-s − 1.45·21-s − 0.776i·23-s − 2.42·25-s − 0.258i·27-s − 0.0164i·29-s + 0.923i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.679 - 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-0.679 - 0.733i$
Analytic conductor: \(32.0359\)
Root analytic conductor: \(5.66003\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4012} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4012,\ (\ :1/2),\ -0.679 - 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.893060759\)
\(L(\frac12)\) \(\approx\) \(1.893060759\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (2.80 + 3.02i)T \)
59 \( 1 + T \)
good3 \( 1 + 2.32iT - 3T^{2} \)
5 \( 1 + 4.13iT - 5T^{2} \)
7 \( 1 + 2.86iT - 7T^{2} \)
11 \( 1 - 0.105iT - 11T^{2} \)
13 \( 1 - 4.79T + 13T^{2} \)
19 \( 1 - 6.25T + 19T^{2} \)
23 \( 1 + 3.72iT - 23T^{2} \)
29 \( 1 + 0.0883iT - 29T^{2} \)
31 \( 1 - 5.14iT - 31T^{2} \)
37 \( 1 - 7.58iT - 37T^{2} \)
41 \( 1 + 9.44iT - 41T^{2} \)
43 \( 1 + 1.19T + 43T^{2} \)
47 \( 1 + 4.55T + 47T^{2} \)
53 \( 1 + 4.31T + 53T^{2} \)
61 \( 1 + 1.14iT - 61T^{2} \)
67 \( 1 - 5.82T + 67T^{2} \)
71 \( 1 + 13.0iT - 71T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 - 5.02T + 83T^{2} \)
89 \( 1 - 8.02T + 89T^{2} \)
97 \( 1 + 18.2iT - 97T^{2} \)
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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

−8.115486743778326634324759646810, −7.22692092689556295378890640351, −6.70878745144739546355053014307, −5.79094614304027581400429236968, −4.97715287391646065987972252576, −4.29629767025786593868429021599, −3.32311413287683852314471719476, −1.86922202743625425087377611175, −1.08893457103619057267010367140, −0.62693400868630221952067565542, 1.88263009298966631049669139509, 2.95643614861324328073972359796, 3.45732157802142558932560679591, 4.12296325812766523857139473872, 5.29524488936068518956347288842, 5.96770022891022631112232473522, 6.47335543992685132602694359088, 7.48608858521870124451274550636, 8.216375268320177481096786146268, 9.237717178495505335614418988610

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