Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.699 + 0.714i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.593i·3-s − 3.87i·5-s − 2.92i·7-s + 2.64·9-s + 4.95i·11-s − 3.80·13-s + 2.29·15-s + (2.88 − 2.94i)17-s + 1.82·19-s + 1.73·21-s − 8.40i·23-s − 10.0·25-s + 3.35i·27-s + 1.64i·29-s − 7.62i·31-s + ⋯
L(s)  = 1  + 0.342i·3-s − 1.73i·5-s − 1.10i·7-s + 0.882·9-s + 1.49i·11-s − 1.05·13-s + 0.593·15-s + (0.699 − 0.714i)17-s + 0.417·19-s + 0.379·21-s − 1.75i·23-s − 2.00·25-s + 0.644i·27-s + 0.305i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.699 + 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.699 + 0.714i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.699 + 0.714i)$
$L(1)$  $\approx$  $1.611405055$
$L(\frac12)$  $\approx$  $1.611405055$
$L(\frac{3}{2})$   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 \( 1 + (-2.88 + 2.94i)T \)
59 \( 1 - T \)
good3 \( 1 - 0.593iT - 3T^{2} \)
5 \( 1 + 3.87iT - 5T^{2} \)
7 \( 1 + 2.92iT - 7T^{2} \)
11 \( 1 - 4.95iT - 11T^{2} \)
13 \( 1 + 3.80T + 13T^{2} \)
19 \( 1 - 1.82T + 19T^{2} \)
23 \( 1 + 8.40iT - 23T^{2} \)
29 \( 1 - 1.64iT - 29T^{2} \)
31 \( 1 + 7.62iT - 31T^{2} \)
37 \( 1 + 0.0884iT - 37T^{2} \)
41 \( 1 + 2.48iT - 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 1.02T + 47T^{2} \)
53 \( 1 - 0.471T + 53T^{2} \)
61 \( 1 - 6.85iT - 61T^{2} \)
67 \( 1 + 12.9T + 67T^{2} \)
71 \( 1 + 8.36iT - 71T^{2} \)
73 \( 1 - 7.15iT - 73T^{2} \)
79 \( 1 + 8.40iT - 79T^{2} \)
83 \( 1 - 2.81T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + 2.93iT - 97T^{2} \)
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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

−8.067639449985279363230039609755, −7.37698344491280718803490984131, −7.07777243172986761795838268352, −5.71968279892840905165035848870, −4.76758483493736316836707629093, −4.53918906599465811465540667500, −3.99883570560042612270793511801, −2.44405171533912485718144679787, −1.36819407317292299403366897622, −0.48183729257110371298599808348, 1.43988703640224016448567008024, 2.53235202496230858516152534092, 3.13804305155213003477616073927, 3.85061102691203002203130227064, 5.27842734724771013507956026955, 5.88344854917726497162554112474, 6.49272186406028540472745572024, 7.36891768519928730407984527416, 7.70429088978262985482652074821, 8.648195561545209161809000591961

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