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Properties

Label 2-4012-17.16-c1-0-19
Degree $2$
Conductor $4012$
Sign $-0.100 + 0.994i$
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  + 2.97i·3-s + 2.24i·5-s + 1.80i·7-s − 5.86·9-s + 3.56i·11-s − 3.28·13-s − 6.68·15-s + (−0.415 + 4.10i)17-s − 3.16·19-s − 5.37·21-s + 3.51i·23-s − 0.0370·25-s − 8.51i·27-s + 4.99i·29-s + 6.66i·31-s + ⋯
L(s)  = 1  + 1.71i·3-s + 1.00i·5-s + 0.682i·7-s − 1.95·9-s + 1.07i·11-s − 0.910·13-s − 1.72·15-s + (−0.100 + 0.994i)17-s − 0.725·19-s − 1.17·21-s + 0.732i·23-s − 0.00741·25-s − 1.63i·27-s + 0.927i·29-s + 1.19i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $-0.100 + 0.994i$
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: \((2,\ 4012,\ (\ :1/2),\ -0.100 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.385807115\)
\(L(\frac12)\) \(\approx\) \(1.385807115\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.415 - 4.10i)T \)
59 \( 1 + T \)
good3 \( 1 - 2.97iT - 3T^{2} \)
5 \( 1 - 2.24iT - 5T^{2} \)
7 \( 1 - 1.80iT - 7T^{2} \)
11 \( 1 - 3.56iT - 11T^{2} \)
13 \( 1 + 3.28T + 13T^{2} \)
19 \( 1 + 3.16T + 19T^{2} \)
23 \( 1 - 3.51iT - 23T^{2} \)
29 \( 1 - 4.99iT - 29T^{2} \)
31 \( 1 - 6.66iT - 31T^{2} \)
37 \( 1 + 4.52iT - 37T^{2} \)
41 \( 1 + 7.64iT - 41T^{2} \)
43 \( 1 - 12.7T + 43T^{2} \)
47 \( 1 - 0.459T + 47T^{2} \)
53 \( 1 - 4.00T + 53T^{2} \)
61 \( 1 - 4.46iT - 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 0.964iT - 71T^{2} \)
73 \( 1 - 1.39iT - 73T^{2} \)
79 \( 1 - 15.9iT - 79T^{2} \)
83 \( 1 + 13.9T + 83T^{2} \)
89 \( 1 - 4.70T + 89T^{2} \)
97 \( 1 + 9.04iT - 97T^{2} \)
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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

−9.039449457630130382459518042071, −8.602980260075299190392101092767, −7.41428636534073767276786062923, −6.83493867749241504299793679110, −5.74934597523041560713622970629, −5.24173431091863301790242060772, −4.34276657326797209851948402259, −3.75585056399799851762836563189, −2.81374913577091969731110844689, −2.11794966572839830852937491567, 0.51162114719795465808903759892, 0.845232524239757691835817273924, 2.15043392606386815461694698236, 2.83853796550642349826049868568, 4.18687839220557725044703323619, 4.96854394853000711388259178762, 5.92819543384598717953091242685, 6.45836062493965641658112876829, 7.32208338272562287097738540557, 7.81380908342040324612366064751

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