Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.912 + 0.408i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.45i·3-s − 3.69i·5-s + 3.33i·7-s + 0.878·9-s + 4.75i·11-s + 2.51·13-s − 5.38·15-s + (−3.76 − 1.68i)17-s − 0.982·19-s + 4.85·21-s + 4.02i·23-s − 8.66·25-s − 5.64i·27-s + 0.375i·29-s + 3.17i·31-s + ⋯
L(s)  = 1  − 0.840i·3-s − 1.65i·5-s + 1.25i·7-s + 0.292·9-s + 1.43i·11-s + 0.697·13-s − 1.39·15-s + (−0.912 − 0.408i)17-s − 0.225·19-s + 1.05·21-s + 0.838i·23-s − 1.73·25-s − 1.08i·27-s + 0.0697i·29-s + 0.569i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.912 + 0.408i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.912 + 0.408i)$
$L(1)$  $\approx$  $1.931199135$
$L(\frac12)$  $\approx$  $1.931199135$
$L(\frac{3}{2})$   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 \( 1 + (3.76 + 1.68i)T \)
59 \( 1 - T \)
good3 \( 1 + 1.45iT - 3T^{2} \)
5 \( 1 + 3.69iT - 5T^{2} \)
7 \( 1 - 3.33iT - 7T^{2} \)
11 \( 1 - 4.75iT - 11T^{2} \)
13 \( 1 - 2.51T + 13T^{2} \)
19 \( 1 + 0.982T + 19T^{2} \)
23 \( 1 - 4.02iT - 23T^{2} \)
29 \( 1 - 0.375iT - 29T^{2} \)
31 \( 1 - 3.17iT - 31T^{2} \)
37 \( 1 + 1.30iT - 37T^{2} \)
41 \( 1 - 7.88iT - 41T^{2} \)
43 \( 1 - 9.82T + 43T^{2} \)
47 \( 1 - 13.2T + 47T^{2} \)
53 \( 1 - 1.89T + 53T^{2} \)
61 \( 1 - 0.836iT - 61T^{2} \)
67 \( 1 - 1.30T + 67T^{2} \)
71 \( 1 - 10.8iT - 71T^{2} \)
73 \( 1 + 0.00541iT - 73T^{2} \)
79 \( 1 + 10.6iT - 79T^{2} \)
83 \( 1 - 7.42T + 83T^{2} \)
89 \( 1 + 4.58T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−8.448293016262634729388638136802, −7.70742405489992174253212152431, −7.02682252754842599825939022738, −6.11682137388286041468926087458, −5.43322586229670254040685506765, −4.66574269679204514499064785785, −4.07817410042761620161415012387, −2.46141907131773455655316549540, −1.81505796783244289359801277336, −0.973206440309139802348748040909, 0.68519926261003701776835020573, 2.28037654573196896791962186363, 3.27521105364934138351578395649, 3.91173405936831256315403186235, 4.30953257365725299393811219239, 5.71036970052559371349247086582, 6.34261743972278132955423760770, 7.00508527093760097499316208025, 7.60708894198013953773511829422, 8.561211069468642448054304721958

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