Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.192 + 0.981i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.274i·3-s + 3.55i·5-s + 0.321i·7-s + 2.92·9-s − 0.790i·11-s − 4.51·13-s + 0.976·15-s + (−0.792 − 4.04i)17-s − 3.21·19-s + 0.0883·21-s + 3.60i·23-s − 7.66·25-s − 1.62i·27-s − 7.35i·29-s − 10.7i·31-s + ⋯
L(s)  = 1  − 0.158i·3-s + 1.59i·5-s + 0.121i·7-s + 0.974·9-s − 0.238i·11-s − 1.25·13-s + 0.252·15-s + (−0.192 − 0.981i)17-s − 0.737·19-s + 0.0192·21-s + 0.751i·23-s − 1.53·25-s − 0.313i·27-s − 1.36i·29-s − 1.93i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.192 + 0.981i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.192 + 0.981i)$
$L(1)$  $\approx$  $1.006135308$
$L(\frac12)$  $\approx$  $1.006135308$
$L(\frac{3}{2})$   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 \( 1 + (0.792 + 4.04i)T \)
59 \( 1 - T \)
good3 \( 1 + 0.274iT - 3T^{2} \)
5 \( 1 - 3.55iT - 5T^{2} \)
7 \( 1 - 0.321iT - 7T^{2} \)
11 \( 1 + 0.790iT - 11T^{2} \)
13 \( 1 + 4.51T + 13T^{2} \)
19 \( 1 + 3.21T + 19T^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 + 7.35iT - 29T^{2} \)
31 \( 1 + 10.7iT - 31T^{2} \)
37 \( 1 + 8.58iT - 37T^{2} \)
41 \( 1 + 7.36iT - 41T^{2} \)
43 \( 1 + 7.18T + 43T^{2} \)
47 \( 1 + 2.61T + 47T^{2} \)
53 \( 1 + 5.58T + 53T^{2} \)
61 \( 1 - 7.98iT - 61T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 - 5.27iT - 73T^{2} \)
79 \( 1 + 1.75iT - 79T^{2} \)
83 \( 1 + 12.3T + 83T^{2} \)
89 \( 1 - 3.48T + 89T^{2} \)
97 \( 1 - 16.6iT - 97T^{2} \)
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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

−7.945027456771467276856164933676, −7.42331094742490185267288365394, −6.96527505034776593759648054823, −6.23900891939177506901006139039, −5.41711362701599071143099611643, −4.33534233750667097485274582887, −3.66060139556577024042714929061, −2.48872567887403125260781137660, −2.16963740092548292134563012852, −0.28890491298086219524321276054, 1.20859681611402389917660458152, 1.88799655112928279118486195377, 3.27372004039585080224721145322, 4.40864860054152655488983618932, 4.72407051666422696774700710640, 5.31852854597421546930013125668, 6.57114284021372226602607745098, 7.04268734131918658839279665931, 8.186826034017790042908160914487, 8.477945355338487796667444218804

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