Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.114 + 0.993i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.34i·3-s + 2.91i·5-s + 3.43i·7-s − 2.47·9-s + 1.22i·11-s − 4.98·13-s − 6.82·15-s + (0.473 − 4.09i)17-s + 3.37·19-s − 8.05·21-s + 0.446i·23-s − 3.49·25-s + 1.21i·27-s + 2.96i·29-s + 1.59i·31-s + ⋯
L(s)  = 1  + 1.35i·3-s + 1.30i·5-s + 1.29i·7-s − 0.826·9-s + 0.368i·11-s − 1.38·13-s − 1.76·15-s + (0.114 − 0.993i)17-s + 0.775·19-s − 1.75·21-s + 0.0931i·23-s − 0.699·25-s + 0.234i·27-s + 0.550i·29-s + 0.285i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.114 + 0.993i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.114 + 0.993i)$
$L(1)$  $\approx$  $1.032429275$
$L(\frac12)$  $\approx$  $1.032429275$
$L(\frac{3}{2})$   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 \( 1 + (-0.473 + 4.09i)T \)
59 \( 1 - T \)
good3 \( 1 - 2.34iT - 3T^{2} \)
5 \( 1 - 2.91iT - 5T^{2} \)
7 \( 1 - 3.43iT - 7T^{2} \)
11 \( 1 - 1.22iT - 11T^{2} \)
13 \( 1 + 4.98T + 13T^{2} \)
19 \( 1 - 3.37T + 19T^{2} \)
23 \( 1 - 0.446iT - 23T^{2} \)
29 \( 1 - 2.96iT - 29T^{2} \)
31 \( 1 - 1.59iT - 31T^{2} \)
37 \( 1 + 3.24iT - 37T^{2} \)
41 \( 1 + 1.29iT - 41T^{2} \)
43 \( 1 + 7.24T + 43T^{2} \)
47 \( 1 - 3.49T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
61 \( 1 + 12.7iT - 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 0.203iT - 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 - 10.4iT - 79T^{2} \)
83 \( 1 - 7.96T + 83T^{2} \)
89 \( 1 + 8.89T + 89T^{2} \)
97 \( 1 + 11.9iT - 97T^{2} \)
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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

−9.380884988870026151057571631208, −8.388465040320185200261692240545, −7.33723356148786190532215475667, −6.92609783791655704984816094896, −5.82303360849581228983793832236, −5.13334516719134090546870417470, −4.63499674166488655196918912784, −3.37081236033604587645449919221, −2.93227262302511761972143244378, −2.11647878130237383702087382228, 0.31001648039262814609921890482, 1.13274927416133524123989210709, 1.85365835184881903450746082637, 3.10859287684333168043756120202, 4.25375226381002016246935521547, 4.83694312309728624469125585612, 5.78882498902242346562613934733, 6.56535735912571882297056134599, 7.37944181993008141903440316412, 7.77555022376465921141734747363

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