Properties

Degree 2
Conductor $ 2^{2} \cdot 17 \cdot 59 $
Sign $0.369 + 0.929i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.257i·3-s + 0.743i·5-s − 3.88i·7-s + 2.93·9-s − 1.06i·11-s + 5.38·13-s + 0.191·15-s + (−1.52 − 3.83i)17-s + 3.07·19-s − 1.00·21-s − 1.73i·23-s + 4.44·25-s − 1.53i·27-s + 4.99i·29-s + 1.18i·31-s + ⋯
L(s)  = 1  − 0.148i·3-s + 0.332i·5-s − 1.47i·7-s + 0.977·9-s − 0.322i·11-s + 1.49·13-s + 0.0495·15-s + (−0.369 − 0.929i)17-s + 0.706·19-s − 0.218·21-s − 0.361i·23-s + 0.889·25-s − 0.294i·27-s + 0.926i·29-s + 0.213i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4012\)    =    \(2^{2} \cdot 17 \cdot 59\)
\( \varepsilon \)  =  $0.369 + 0.929i$
motivic weight  =  \(1\)
character  :  $\chi_{4012} (237, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4012,\ (\ :1/2),\ 0.369 + 0.929i)$
$L(1)$  $\approx$  $2.308424857$
$L(\frac12)$  $\approx$  $2.308424857$
$L(\frac{3}{2})$   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 \( 1 + (1.52 + 3.83i)T \)
59 \( 1 - T \)
good3 \( 1 + 0.257iT - 3T^{2} \)
5 \( 1 - 0.743iT - 5T^{2} \)
7 \( 1 + 3.88iT - 7T^{2} \)
11 \( 1 + 1.06iT - 11T^{2} \)
13 \( 1 - 5.38T + 13T^{2} \)
19 \( 1 - 3.07T + 19T^{2} \)
23 \( 1 + 1.73iT - 23T^{2} \)
29 \( 1 - 4.99iT - 29T^{2} \)
31 \( 1 - 1.18iT - 31T^{2} \)
37 \( 1 - 0.759iT - 37T^{2} \)
41 \( 1 + 3.00iT - 41T^{2} \)
43 \( 1 + 4.92T + 43T^{2} \)
47 \( 1 - 5.80T + 47T^{2} \)
53 \( 1 + 0.194T + 53T^{2} \)
61 \( 1 + 4.03iT - 61T^{2} \)
67 \( 1 + 7.58T + 67T^{2} \)
71 \( 1 - 6.78iT - 71T^{2} \)
73 \( 1 - 13.3iT - 73T^{2} \)
79 \( 1 + 15.2iT - 79T^{2} \)
83 \( 1 + 8.30T + 83T^{2} \)
89 \( 1 - 8.68T + 89T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−8.263301233226503098647119158133, −7.30168474352043420065980323713, −7.01033830134824941093933706755, −6.34930167599163964641797491386, −5.24972368223124960464418285627, −4.38558237429452837155079937661, −3.71623028046564865248141404274, −2.96235223002059938764491018352, −1.47325356510373469849575620484, −0.78450696811166414407732991533, 1.23036771033982901424634631799, 2.04820491389050508710260425392, 3.16955499457053019364564634662, 4.05561262772171569228998193707, 4.82152340845471014856879700232, 5.72277507726710935016824944779, 6.20432650834747722962960239820, 7.08878354128958995933521768221, 8.053293451539312501422741159703, 8.628159500581464762636036646203

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