Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.620 - 0.784i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.123i·3-s + 0.454i·5-s + 3.56i·7-s + 2.98·9-s + 2.55i·11-s + 1.05·13-s − 0.0561·15-s + (2.55 + 3.23i)17-s − 5.45·19-s − 0.439·21-s + 5.82i·23-s + 4.79·25-s + 0.739i·27-s + 10.2i·29-s − 0.560i·31-s + ⋯
L(s)  = 1  + 0.0713i·3-s + 0.203i·5-s + 1.34i·7-s + 0.994·9-s + 0.771i·11-s + 0.293·13-s − 0.0144·15-s + (0.620 + 0.784i)17-s − 1.25·19-s − 0.0959·21-s + 1.21i·23-s + 0.958·25-s + 0.142i·27-s + 1.91i·29-s − 0.100i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.620 - 0.784i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.620 - 0.784i)$
$L(1)$  $\approx$  $1.803188823$
$L(\frac12)$  $\approx$  $1.803188823$
$L(\frac{3}{2})$   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 \( 1 + (-2.55 - 3.23i)T \)
59 \( 1 - T \)
good3 \( 1 - 0.123iT - 3T^{2} \)
5 \( 1 - 0.454iT - 5T^{2} \)
7 \( 1 - 3.56iT - 7T^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
19 \( 1 + 5.45T + 19T^{2} \)
23 \( 1 - 5.82iT - 23T^{2} \)
29 \( 1 - 10.2iT - 29T^{2} \)
31 \( 1 + 0.560iT - 31T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + 6.36iT - 41T^{2} \)
43 \( 1 + 9.85T + 43T^{2} \)
47 \( 1 + 4.05T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
61 \( 1 + 6.58iT - 61T^{2} \)
67 \( 1 + 2.95T + 67T^{2} \)
71 \( 1 - 4.41iT - 71T^{2} \)
73 \( 1 - 7.04iT - 73T^{2} \)
79 \( 1 + 7.96iT - 79T^{2} \)
83 \( 1 - 11.8T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
97 \( 1 - 9.68iT - 97T^{2} \)
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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

−8.847394523782149011810107404660, −8.026672958142214401057691368155, −7.11874041608160886149800587120, −6.63044587739384946032748313058, −5.61452075295182275042145942861, −5.12053103242301359617739374942, −4.08472524879986466676297492709, −3.34000181359291149742845063304, −2.16919194788448652062118871058, −1.53217921712079149143355568648, 0.54426271011784927802388482193, 1.32834044139297136968475621198, 2.66613863217163943939863905130, 3.67683782445257156433443248253, 4.41144846996754022329020931913, 4.92623961058449945752623866969, 6.27848199335378697377896640703, 6.63789943910867378389911671371, 7.46007490580040828610207980407, 8.191016857682479069029997730657

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