Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $-0.511 + 0.859i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.27i·3-s + 2.55i·5-s − 1.00i·7-s − 2.18·9-s + 4.20i·11-s − 6.65·13-s + 5.82·15-s + (2.10 − 3.54i)17-s + 1.96·19-s − 2.28·21-s − 7.06i·23-s − 1.54·25-s − 1.86i·27-s + 2.50i·29-s + 7.19i·31-s + ⋯
L(s)  = 1  − 1.31i·3-s + 1.14i·5-s − 0.379i·7-s − 0.727·9-s + 1.26i·11-s − 1.84·13-s + 1.50·15-s + (0.511 − 0.859i)17-s + 0.450·19-s − 0.499·21-s − 1.47i·23-s − 0.309·25-s − 0.358i·27-s + 0.464i·29-s + 1.29i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $-0.511 + 0.859i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ -0.511 + 0.859i)$
$L(1)$  $\approx$  $1.210722812$
$L(\frac12)$  $\approx$  $1.210722812$
$L(\frac{3}{2})$   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 \( 1 + (-2.10 + 3.54i)T \)
59 \( 1 - T \)
good3 \( 1 + 2.27iT - 3T^{2} \)
5 \( 1 - 2.55iT - 5T^{2} \)
7 \( 1 + 1.00iT - 7T^{2} \)
11 \( 1 - 4.20iT - 11T^{2} \)
13 \( 1 + 6.65T + 13T^{2} \)
19 \( 1 - 1.96T + 19T^{2} \)
23 \( 1 + 7.06iT - 23T^{2} \)
29 \( 1 - 2.50iT - 29T^{2} \)
31 \( 1 - 7.19iT - 31T^{2} \)
37 \( 1 - 5.03iT - 37T^{2} \)
41 \( 1 + 11.6iT - 41T^{2} \)
43 \( 1 - 2.35T + 43T^{2} \)
47 \( 1 + 9.91T + 47T^{2} \)
53 \( 1 - 0.260T + 53T^{2} \)
61 \( 1 + 10.4iT - 61T^{2} \)
67 \( 1 - 3.87T + 67T^{2} \)
71 \( 1 - 5.64iT - 71T^{2} \)
73 \( 1 + 8.82iT - 73T^{2} \)
79 \( 1 + 13.6iT - 79T^{2} \)
83 \( 1 - 2.15T + 83T^{2} \)
89 \( 1 - 4.88T + 89T^{2} \)
97 \( 1 + 12.2iT - 97T^{2} \)
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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

−7.76251419057198099330443866763, −7.19890682559577117497582117170, −7.05236767634348733518875863830, −6.42231638927380243221700788839, −5.15780372064458561746372565767, −4.57900507903427989081140435097, −3.18324911116128212404143615089, −2.50597514567892720071766701694, −1.79091659840656295209464416195, −0.37951600855096401177276261124, 1.06819940268752361039243213948, 2.48054942096953225928705782070, 3.49423613674064536583679836031, 4.17229189706804644614354898448, 5.06696874356507171157901595994, 5.38478936992976335224114734848, 6.15139306735432773589336666943, 7.52735109241823452545275492140, 8.066663325408308920700299678872, 8.891579503158709359295223121914

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