Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.994 - 0.108i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.405i·3-s + 1.20i·5-s − 0.122i·7-s + 2.83·9-s + 1.65i·11-s − 1.21·13-s + 0.488·15-s + (−4.09 + 0.446i)17-s + 5.49·19-s − 0.0496·21-s − 1.29i·23-s + 3.54·25-s − 2.36i·27-s − 9.31i·29-s − 6.06i·31-s + ⋯
L(s)  = 1  − 0.233i·3-s + 0.538i·5-s − 0.0462i·7-s + 0.945·9-s + 0.498i·11-s − 0.337·13-s + 0.126·15-s + (−0.994 + 0.108i)17-s + 1.26·19-s − 0.0108·21-s − 0.269i·23-s + 0.709·25-s − 0.454i·27-s − 1.72i·29-s − 1.08i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.994 - 0.108i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.994 - 0.108i)$
$L(1)$  $\approx$  $2.063342975$
$L(\frac12)$  $\approx$  $2.063342975$
$L(\frac{3}{2})$   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 \( 1 + (4.09 - 0.446i)T \)
59 \( 1 - T \)
good3 \( 1 + 0.405iT - 3T^{2} \)
5 \( 1 - 1.20iT - 5T^{2} \)
7 \( 1 + 0.122iT - 7T^{2} \)
11 \( 1 - 1.65iT - 11T^{2} \)
13 \( 1 + 1.21T + 13T^{2} \)
19 \( 1 - 5.49T + 19T^{2} \)
23 \( 1 + 1.29iT - 23T^{2} \)
29 \( 1 + 9.31iT - 29T^{2} \)
31 \( 1 + 6.06iT - 31T^{2} \)
37 \( 1 - 4.07iT - 37T^{2} \)
41 \( 1 - 1.98iT - 41T^{2} \)
43 \( 1 - 7.44T + 43T^{2} \)
47 \( 1 + 10.1T + 47T^{2} \)
53 \( 1 - 4.71T + 53T^{2} \)
61 \( 1 - 3.86iT - 61T^{2} \)
67 \( 1 - 2.82T + 67T^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + 0.917iT - 79T^{2} \)
83 \( 1 - 12.7T + 83T^{2} \)
89 \( 1 + 2.24T + 89T^{2} \)
97 \( 1 + 7.92iT - 97T^{2} \)
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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

−8.310527983973165346602173296185, −7.58681994783303098253091732845, −7.02227648630797534247322642798, −6.45540835181264792107260901428, −5.53031230478764788026741578549, −4.52693977025803491650515308216, −4.02161268257019146879878187593, −2.79398876239553516663910194757, −2.09665753356158987836180234228, −0.854785282133091313732816738170, 0.841086007343673728656083352250, 1.82368785904685883874723491503, 3.07650724139413755186994393918, 3.81956612161386746582646232450, 4.94756991084700169980325135003, 5.07698757126377604428298127494, 6.27012217514218614859013690239, 7.09321705918061736925479074617, 7.56544583909486396342494426710, 8.659862785577928638021377055377

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