Properties

Label 2-4012-17.16-c1-0-28
Degree $2$
Conductor $4012$
Sign $0.0272 - 0.999i$
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  + 2.18i·3-s + 0.351i·5-s − 3.15i·7-s − 1.75·9-s + 3.05i·11-s − 2.45·13-s − 0.767·15-s + (0.112 − 4.12i)17-s + 2.18·19-s + 6.88·21-s + 0.185i·23-s + 4.87·25-s + 2.71i·27-s + 4.16i·29-s − 6.26i·31-s + ⋯
L(s)  = 1  + 1.25i·3-s + 0.157i·5-s − 1.19i·7-s − 0.585·9-s + 0.922i·11-s − 0.680·13-s − 0.198·15-s + (0.0272 − 0.999i)17-s + 0.500·19-s + 1.50·21-s + 0.0387i·23-s + 0.975·25-s + 0.521i·27-s + 0.774i·29-s − 1.12i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0272 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $0.0272 - 0.999i$
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: \((2,\ 4012,\ (\ :1/2),\ 0.0272 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.772732322\)
\(L(\frac12)\) \(\approx\) \(1.772732322\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.112 + 4.12i)T \)
59 \( 1 + T \)
good3 \( 1 - 2.18iT - 3T^{2} \)
5 \( 1 - 0.351iT - 5T^{2} \)
7 \( 1 + 3.15iT - 7T^{2} \)
11 \( 1 - 3.05iT - 11T^{2} \)
13 \( 1 + 2.45T + 13T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 - 0.185iT - 23T^{2} \)
29 \( 1 - 4.16iT - 29T^{2} \)
31 \( 1 + 6.26iT - 31T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + 6.35iT - 41T^{2} \)
43 \( 1 + 2.08T + 43T^{2} \)
47 \( 1 - 0.937T + 47T^{2} \)
53 \( 1 - 9.12T + 53T^{2} \)
61 \( 1 + 1.52iT - 61T^{2} \)
67 \( 1 - 8.22T + 67T^{2} \)
71 \( 1 - 11.2iT - 71T^{2} \)
73 \( 1 + 11.1iT - 73T^{2} \)
79 \( 1 - 14.0iT - 79T^{2} \)
83 \( 1 - 3.09T + 83T^{2} \)
89 \( 1 - 6.46T + 89T^{2} \)
97 \( 1 - 16.4iT - 97T^{2} \)
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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

−8.875910203846384204441595037871, −7.73817220009114881324962223977, −7.18708503317717269856930834630, −6.62629546987070922862441382901, −5.18007834395745969626396610838, −4.89415698410428894088348435831, −4.10248506734983037235732358091, −3.41726273067407058145996075672, −2.43068877468862823155282746268, −0.951810795980915826763539971551, 0.63304873000287577761371212379, 1.75666354860780592423843462448, 2.51375084553376581937917489089, 3.37508201161867812336793929561, 4.61198960708531111994936612690, 5.61923681475499698924626818391, 5.99050099645119848402459181403, 6.84707135114915656533225874693, 7.50566542101952084510318171620, 8.343377990665935254105877648891

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