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Properties

Label 2-4012-17.16-c1-0-31
Degree $2$
Conductor $4012$
Sign $-0.252 - 0.967i$
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  + 1.13i·3-s + 2.09i·5-s − 2.71i·7-s + 1.71·9-s + 3.86i·11-s − 2.19·13-s − 2.37·15-s + (1.04 + 3.98i)17-s + 7.57·19-s + 3.07·21-s − 3.94i·23-s + 0.603·25-s + 5.34i·27-s + 2.64i·29-s + 4.90i·31-s + ⋯
L(s)  = 1  + 0.654i·3-s + 0.937i·5-s − 1.02i·7-s + 0.572·9-s + 1.16i·11-s − 0.608·13-s − 0.613·15-s + (0.252 + 0.967i)17-s + 1.73·19-s + 0.669·21-s − 0.821i·23-s + 0.120·25-s + 1.02i·27-s + 0.491i·29-s + 0.880i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-0.252 - 0.967i$
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: \((2,\ 4012,\ (\ :1/2),\ -0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.009140297\)
\(L(\frac12)\) \(\approx\) \(2.009140297\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.04 - 3.98i)T \)
59 \( 1 - T \)
good3 \( 1 - 1.13iT - 3T^{2} \)
5 \( 1 - 2.09iT - 5T^{2} \)
7 \( 1 + 2.71iT - 7T^{2} \)
11 \( 1 - 3.86iT - 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
19 \( 1 - 7.57T + 19T^{2} \)
23 \( 1 + 3.94iT - 23T^{2} \)
29 \( 1 - 2.64iT - 29T^{2} \)
31 \( 1 - 4.90iT - 31T^{2} \)
37 \( 1 + 11.3iT - 37T^{2} \)
41 \( 1 + 4.94iT - 41T^{2} \)
43 \( 1 + 5.42T + 43T^{2} \)
47 \( 1 - 6.03T + 47T^{2} \)
53 \( 1 + 1.40T + 53T^{2} \)
61 \( 1 - 13.3iT - 61T^{2} \)
67 \( 1 - 1.98T + 67T^{2} \)
71 \( 1 - 4.42iT - 71T^{2} \)
73 \( 1 + 1.70iT - 73T^{2} \)
79 \( 1 - 4.77iT - 79T^{2} \)
83 \( 1 - 9.20T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 - 14.2iT - 97T^{2} \)
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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

−8.779357859409272993904401895539, −7.55400846731536598822161834050, −7.22023709919276602965279302262, −6.79336922264861429409606452610, −5.55425537282578870729775927197, −4.78735991495933105710136106719, −4.01795765514467711322402531760, −3.43947579995335641400045544121, −2.36153618202169901682200463949, −1.18608429562315960849453801644, 0.66488539385863603208511549114, 1.48176028474446509081075650660, 2.64886719746522558069144025008, 3.40831046707652419700677490260, 4.73400696170491132636015235874, 5.24422990152635717090515661439, 5.93694330759916094106748546129, 6.78551015304130667063886059812, 7.70575552100011916706766323133, 8.077931688518768798362721024609

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