Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.0437 - 0.999i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.50i·3-s + 4.31i·5-s − 4.93i·7-s + 0.739·9-s + 1.58i·11-s − 0.143·13-s + 6.48·15-s + (−0.180 + 4.11i)17-s + 0.296·19-s − 7.41·21-s + 6.24i·23-s − 13.5·25-s − 5.62i·27-s + 6.17i·29-s − 4.07i·31-s + ⋯
L(s)  = 1  − 0.868i·3-s + 1.92i·5-s − 1.86i·7-s + 0.246·9-s + 0.477i·11-s − 0.0398·13-s + 1.67·15-s + (−0.0437 + 0.999i)17-s + 0.0680·19-s − 1.61·21-s + 1.30i·23-s − 2.71·25-s − 1.08i·27-s + 1.14i·29-s − 0.732i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0437 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.0437 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.0437 - 0.999i)$
$L(1)$  $\approx$  $1.277511486$
$L(\frac12)$  $\approx$  $1.277511486$
$L(\frac{3}{2})$   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 \( 1 + (0.180 - 4.11i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.50iT - 3T^{2} \)
5 \( 1 - 4.31iT - 5T^{2} \)
7 \( 1 + 4.93iT - 7T^{2} \)
11 \( 1 - 1.58iT - 11T^{2} \)
13 \( 1 + 0.143T + 13T^{2} \)
19 \( 1 - 0.296T + 19T^{2} \)
23 \( 1 - 6.24iT - 23T^{2} \)
29 \( 1 - 6.17iT - 29T^{2} \)
31 \( 1 + 4.07iT - 31T^{2} \)
37 \( 1 - 7.76iT - 37T^{2} \)
41 \( 1 - 1.59iT - 41T^{2} \)
43 \( 1 + 6.67T + 43T^{2} \)
47 \( 1 + 11.8T + 47T^{2} \)
53 \( 1 - 6.38T + 53T^{2} \)
61 \( 1 + 0.241iT - 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 - 1.01iT - 71T^{2} \)
73 \( 1 - 0.169iT - 73T^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + 5.97T + 83T^{2} \)
89 \( 1 - 3.73T + 89T^{2} \)
97 \( 1 - 4.04iT - 97T^{2} \)
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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

−8.137589395481108211903249687535, −7.64235644991130226311626890550, −7.06714153349816038518976871579, −6.73269704366558591089541230316, −6.12010575366843448295823570024, −4.74799910308078436617333941085, −3.70718391391548743028570150418, −3.37094443785990029765741152057, −2.06858053746216774467538465848, −1.29095597674915153552697941413, 0.36727481691341190935711618698, 1.72814496264692897592035193310, 2.68696742513477381582811300675, 3.86497866748564914982294613503, 4.70002737406065498731703008568, 5.19494718231353881573551798133, 5.64555345471794419153406303055, 6.62013965691821426643833391821, 7.932840476015882697000283683377, 8.625057886685036709836474774525

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