Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.775 - 0.631i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.57i·3-s + 2.31i·5-s + 3.71i·7-s − 3.61·9-s + 5.32i·11-s − 2.64·13-s + 5.96·15-s + (3.19 + 2.60i)17-s − 4.86·19-s + 9.55·21-s − 0.402i·23-s − 0.378·25-s + 1.58i·27-s − 7.10i·29-s − 5.18i·31-s + ⋯
L(s)  = 1  − 1.48i·3-s + 1.03i·5-s + 1.40i·7-s − 1.20·9-s + 1.60i·11-s − 0.732·13-s + 1.54·15-s + (0.775 + 0.631i)17-s − 1.11·19-s + 2.08·21-s − 0.0839i·23-s − 0.0756·25-s + 0.305i·27-s − 1.31i·29-s − 0.931i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.775 - 0.631i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.775 - 0.631i)$
$L(1)$  $\approx$  $0.7208529883$
$L(\frac12)$  $\approx$  $0.7208529883$
$L(\frac{3}{2})$   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 \( 1 + (-3.19 - 2.60i)T \)
59 \( 1 - T \)
good3 \( 1 + 2.57iT - 3T^{2} \)
5 \( 1 - 2.31iT - 5T^{2} \)
7 \( 1 - 3.71iT - 7T^{2} \)
11 \( 1 - 5.32iT - 11T^{2} \)
13 \( 1 + 2.64T + 13T^{2} \)
19 \( 1 + 4.86T + 19T^{2} \)
23 \( 1 + 0.402iT - 23T^{2} \)
29 \( 1 + 7.10iT - 29T^{2} \)
31 \( 1 + 5.18iT - 31T^{2} \)
37 \( 1 + 0.811iT - 37T^{2} \)
41 \( 1 - 11.0iT - 41T^{2} \)
43 \( 1 + 6.22T + 43T^{2} \)
47 \( 1 - 4.15T + 47T^{2} \)
53 \( 1 + 9.41T + 53T^{2} \)
61 \( 1 + 9.06iT - 61T^{2} \)
67 \( 1 - 14.3T + 67T^{2} \)
71 \( 1 - 13.3iT - 71T^{2} \)
73 \( 1 + 13.8iT - 73T^{2} \)
79 \( 1 - 13.1iT - 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 + 14.8T + 89T^{2} \)
97 \( 1 - 1.69iT - 97T^{2} \)
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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

−8.323320608716348847079409509409, −7.971735778355224833581992683079, −7.19574601836079918837014994958, −6.54748139746669817379968756371, −6.13925177547615598752962230209, −5.20542648872628984091198718320, −4.16875605555501004439904072804, −2.74662630397077128794728682722, −2.34735383706785399021066903285, −1.65948438887079862812143531345, 0.20770246517811737990256383002, 1.24085789705403287451775139409, 3.02181373597874630501229529835, 3.68228289313613409416194781079, 4.35200556477962400836238201390, 5.08295680508401538563132011175, 5.51226306768231809144765461801, 6.71341548973658293112479604384, 7.50449733735892081113458569891, 8.612160331126864188902242160204

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