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Properties

Label 2-4012-17.16-c1-0-44
Degree $2$
Conductor $4012$
Sign $0.985 - 0.166i$
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  + 0.409i·3-s − 0.188i·5-s + 3.66i·7-s + 2.83·9-s − 2.00i·11-s − 4.06·13-s + 0.0770·15-s + (4.06 − 0.688i)17-s + 0.775·19-s − 1.49·21-s − 7.50i·23-s + 4.96·25-s + 2.38i·27-s + 1.46i·29-s − 6.48i·31-s + ⋯
L(s)  = 1  + 0.236i·3-s − 0.0841i·5-s + 1.38i·7-s + 0.944·9-s − 0.604i·11-s − 1.12·13-s + 0.0199·15-s + (0.985 − 0.166i)17-s + 0.177·19-s − 0.327·21-s − 1.56i·23-s + 0.992·25-s + 0.459i·27-s + 0.272i·29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
Sign: $0.985 - 0.166i$
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: \((2,\ 4012,\ (\ :1/2),\ 0.985 - 0.166i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.053890047\)
\(L(\frac12)\) \(\approx\) \(2.053890047\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.06 + 0.688i)T \)
59 \( 1 + T \)
good3 \( 1 - 0.409iT - 3T^{2} \)
5 \( 1 + 0.188iT - 5T^{2} \)
7 \( 1 - 3.66iT - 7T^{2} \)
11 \( 1 + 2.00iT - 11T^{2} \)
13 \( 1 + 4.06T + 13T^{2} \)
19 \( 1 - 0.775T + 19T^{2} \)
23 \( 1 + 7.50iT - 23T^{2} \)
29 \( 1 - 1.46iT - 29T^{2} \)
31 \( 1 + 6.48iT - 31T^{2} \)
37 \( 1 + 8.93iT - 37T^{2} \)
41 \( 1 - 0.206iT - 41T^{2} \)
43 \( 1 - 4.17T + 43T^{2} \)
47 \( 1 - 10.5T + 47T^{2} \)
53 \( 1 + 9.50T + 53T^{2} \)
61 \( 1 - 5.69iT - 61T^{2} \)
67 \( 1 - 13.1T + 67T^{2} \)
71 \( 1 - 11.7iT - 71T^{2} \)
73 \( 1 + 4.40iT - 73T^{2} \)
79 \( 1 + 0.682iT - 79T^{2} \)
83 \( 1 - 7.95T + 83T^{2} \)
89 \( 1 + 1.45T + 89T^{2} \)
97 \( 1 + 7.60iT - 97T^{2} \)
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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

−8.577407771505644184269725992951, −7.73216383745190422583936168095, −7.07294441280143436242925040710, −6.13275968614830870105079217837, −5.45024515822848865154785203233, −4.80038227692901089752541021124, −3.92808666891900418671383520753, −2.80094947017140944709716328474, −2.22396161637110521982757509280, −0.77781634016201917086222974182, 0.935066525401149216601598133045, 1.71231532648538131050171479205, 3.03295663665160316174593141603, 3.85198362050832351234792283016, 4.66866868612332615403257095343, 5.25280577726601959531140738579, 6.50268430407395700705477445543, 7.15978981800474416759590016373, 7.48941381669655996556441768958, 8.128527811964278664892134024003

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