Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.842 - 0.539i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.81i·3-s − 1.77i·5-s + 1.01i·7-s − 4.92·9-s − 1.98i·11-s + 5.96·13-s + 4.99·15-s + (−3.47 + 2.22i)17-s + 1.12·19-s − 2.85·21-s − 3.54i·23-s + 1.85·25-s − 5.42i·27-s + 3.47i·29-s − 9.40i·31-s + ⋯
L(s)  = 1  + 1.62i·3-s − 0.793i·5-s + 0.382i·7-s − 1.64·9-s − 0.597i·11-s + 1.65·13-s + 1.28·15-s + (−0.842 + 0.539i)17-s + 0.258·19-s − 0.622·21-s − 0.738i·23-s + 0.370·25-s − 1.04i·27-s + 0.645i·29-s − 1.68i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.842 - 0.539i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.842 - 0.539i)$
$L(1)$  $\approx$  $1.904636254$
$L(\frac12)$  $\approx$  $1.904636254$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;17,\;59\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;17,\;59\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (3.47 - 2.22i)T \)
59 \( 1 - T \)
good3 \( 1 - 2.81iT - 3T^{2} \)
5 \( 1 + 1.77iT - 5T^{2} \)
7 \( 1 - 1.01iT - 7T^{2} \)
11 \( 1 + 1.98iT - 11T^{2} \)
13 \( 1 - 5.96T + 13T^{2} \)
19 \( 1 - 1.12T + 19T^{2} \)
23 \( 1 + 3.54iT - 23T^{2} \)
29 \( 1 - 3.47iT - 29T^{2} \)
31 \( 1 + 9.40iT - 31T^{2} \)
37 \( 1 + 4.75iT - 37T^{2} \)
41 \( 1 + 10.0iT - 41T^{2} \)
43 \( 1 + 2.36T + 43T^{2} \)
47 \( 1 + 5.00T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
61 \( 1 + 7.35iT - 61T^{2} \)
67 \( 1 - 9.95T + 67T^{2} \)
71 \( 1 + 1.41iT - 71T^{2} \)
73 \( 1 - 13.9iT - 73T^{2} \)
79 \( 1 + 1.05iT - 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + 10.4iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.556677264684644790082808601191, −8.428119944980499195923153553966, −6.94501700095657045902560825476, −5.85403472606905874270471450728, −5.58615534224275423020804327033, −4.60533097112735564269209176921, −3.98038708953789368166486913500, −3.41661495323062711101720613507, −2.19713671991718761117693011239, −0.69225803796983201781551175190, 0.962458549213228929918093416170, 1.72892790435761139961811952917, 2.74997374464709983126727504021, 3.48477225263720395113391351857, 4.63236576433520516613719856514, 5.74107900230731531389597129476, 6.48792378225729521022138446746, 6.91519807569765027875480083879, 7.40418254195725642405161314800, 8.297749481030067730540112015589

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