Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.952 - 0.304i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.74i·3-s − 1.20i·5-s + 3.93i·7-s − 4.53·9-s + 1.10i·11-s − 0.0851·13-s + 3.30·15-s + (3.92 + 1.25i)17-s + 7.38·19-s − 10.8·21-s − 5.67i·23-s + 3.55·25-s − 4.21i·27-s + 5.59i·29-s + 5.38i·31-s + ⋯
L(s)  = 1  + 1.58i·3-s − 0.537i·5-s + 1.48i·7-s − 1.51·9-s + 0.333i·11-s − 0.0236·13-s + 0.852·15-s + (0.952 + 0.304i)17-s + 1.69·19-s − 2.35·21-s − 1.18i·23-s + 0.710·25-s − 0.810i·27-s + 1.03i·29-s + 0.967i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.952 - 0.304i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.952 - 0.304i)$
$L(1)$  $\approx$  $1.938700455$
$L(\frac12)$  $\approx$  $1.938700455$
$L(\frac{3}{2})$   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 \( 1 + (-3.92 - 1.25i)T \)
59 \( 1 - T \)
good3 \( 1 - 2.74iT - 3T^{2} \)
5 \( 1 + 1.20iT - 5T^{2} \)
7 \( 1 - 3.93iT - 7T^{2} \)
11 \( 1 - 1.10iT - 11T^{2} \)
13 \( 1 + 0.0851T + 13T^{2} \)
19 \( 1 - 7.38T + 19T^{2} \)
23 \( 1 + 5.67iT - 23T^{2} \)
29 \( 1 - 5.59iT - 29T^{2} \)
31 \( 1 - 5.38iT - 31T^{2} \)
37 \( 1 - 7.96iT - 37T^{2} \)
41 \( 1 - 7.91iT - 41T^{2} \)
43 \( 1 - 4.18T + 43T^{2} \)
47 \( 1 - 4.24T + 47T^{2} \)
53 \( 1 - 3.82T + 53T^{2} \)
61 \( 1 - 2.00iT - 61T^{2} \)
67 \( 1 + 8.91T + 67T^{2} \)
71 \( 1 + 8.59iT - 71T^{2} \)
73 \( 1 + 7.38iT - 73T^{2} \)
79 \( 1 + 13.5iT - 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 15.6iT - 97T^{2} \)
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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

−8.849224313013710411328751746263, −8.472045674805936488268251696504, −7.44910371300335462220169450535, −6.28917390688059246741202982644, −5.46534741082180544532695007837, −5.03785251800281797640878463491, −4.46344406112291486020534365434, −3.22347285788880977348253192684, −2.87833401575047337069929559834, −1.34652419433164178455093540493, 0.65672289987713038109273314674, 1.21920014560868358473894996785, 2.44072081471783622073281966713, 3.32973207474663907735496339641, 4.09326508401528238038797693404, 5.50546535485963749130552173475, 5.93706549667870383847562118485, 7.12035634497900233421046930738, 7.31627086533610321909771686954, 7.62774936622977192072518754606

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