Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.990 - 0.138i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.18i·3-s − 1.45i·5-s − 1.23i·7-s − 7.13·9-s + 5.92i·11-s − 3.31·13-s − 4.64·15-s + (−4.08 + 0.571i)17-s − 2.44·19-s − 3.92·21-s + 4.79i·23-s + 2.87·25-s + 13.1i·27-s − 2.32i·29-s − 1.94i·31-s + ⋯
L(s)  = 1  − 1.83i·3-s − 0.652i·5-s − 0.465i·7-s − 2.37·9-s + 1.78i·11-s − 0.918·13-s − 1.19·15-s + (−0.990 + 0.138i)17-s − 0.562·19-s − 0.856·21-s + 0.998i·23-s + 0.574·25-s + 2.53i·27-s − 0.430i·29-s − 0.349i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.990 - 0.138i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.990 - 0.138i)$
$L(1)$  $\approx$  $0.6526038219$
$L(\frac12)$  $\approx$  $0.6526038219$
$L(\frac{3}{2})$   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 \( 1 + (4.08 - 0.571i)T \)
59 \( 1 - T \)
good3 \( 1 + 3.18iT - 3T^{2} \)
5 \( 1 + 1.45iT - 5T^{2} \)
7 \( 1 + 1.23iT - 7T^{2} \)
11 \( 1 - 5.92iT - 11T^{2} \)
13 \( 1 + 3.31T + 13T^{2} \)
19 \( 1 + 2.44T + 19T^{2} \)
23 \( 1 - 4.79iT - 23T^{2} \)
29 \( 1 + 2.32iT - 29T^{2} \)
31 \( 1 + 1.94iT - 31T^{2} \)
37 \( 1 + 3.37iT - 37T^{2} \)
41 \( 1 + 8.19iT - 41T^{2} \)
43 \( 1 + 7.42T + 43T^{2} \)
47 \( 1 - 2.88T + 47T^{2} \)
53 \( 1 - 1.48T + 53T^{2} \)
61 \( 1 - 11.0iT - 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 8.97iT - 71T^{2} \)
73 \( 1 - 16.1iT - 73T^{2} \)
79 \( 1 - 7.60iT - 79T^{2} \)
83 \( 1 - 9.36T + 83T^{2} \)
89 \( 1 - 4.54T + 89T^{2} \)
97 \( 1 - 0.608iT - 97T^{2} \)
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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

−8.302977944208937606022637904248, −7.43466335881691203558958442128, −7.16840941696640430953302681173, −6.58523553880272046902119939747, −5.55893846310916501352714231638, −4.82160040223330681158543135244, −3.93623374723104003327418071125, −2.36120774113100935973574020165, −2.05534888398814469214787519014, −0.973005040135631750577136282306, 0.20858071889671380185052270413, 2.49524849832570122296425295292, 3.04535181207552708398356263461, 3.77131996579419648878427213966, 4.75395151353130561036862440288, 5.18075578443863058967310397004, 6.19309747781268291243229807402, 6.64357070880446099787641590705, 8.047292859236832927933354145307, 8.708431484009239621788097549665

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