Properties

Degree 2
Conductor $ 2 \cdot 2017 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.51·3-s + 4-s + 3.18·5-s − 2.51·6-s + 3.34·7-s + 8-s + 3.33·9-s + 3.18·10-s + 4.39·11-s − 2.51·12-s − 1.65·13-s + 3.34·14-s − 8.01·15-s + 16-s + 0.437·17-s + 3.33·18-s + 6.06·19-s + 3.18·20-s − 8.42·21-s + 4.39·22-s − 0.910·23-s − 2.51·24-s + 5.13·25-s − 1.65·26-s − 0.853·27-s + 3.34·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.45·3-s + 0.5·4-s + 1.42·5-s − 1.02·6-s + 1.26·7-s + 0.353·8-s + 1.11·9-s + 1.00·10-s + 1.32·11-s − 0.726·12-s − 0.459·13-s + 0.893·14-s − 2.06·15-s + 0.250·16-s + 0.106·17-s + 0.787·18-s + 1.39·19-s + 0.711·20-s − 1.83·21-s + 0.937·22-s − 0.189·23-s − 0.513·24-s + 1.02·25-s − 0.324·26-s − 0.164·27-s + 0.632·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4034\)    =    \(2 \cdot 2017\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4034} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4034,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.322803580$
$L(\frac12)$  $\approx$  $3.322803580$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;2017\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;2017\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 - T \)
2017 \( 1 + T \)
good3 \( 1 + 2.51T + 3T^{2} \)
5 \( 1 - 3.18T + 5T^{2} \)
7 \( 1 - 3.34T + 7T^{2} \)
11 \( 1 - 4.39T + 11T^{2} \)
13 \( 1 + 1.65T + 13T^{2} \)
17 \( 1 - 0.437T + 17T^{2} \)
19 \( 1 - 6.06T + 19T^{2} \)
23 \( 1 + 0.910T + 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 - 7.69T + 31T^{2} \)
37 \( 1 + 2.14T + 37T^{2} \)
41 \( 1 + 1.40T + 41T^{2} \)
43 \( 1 + 7.88T + 43T^{2} \)
47 \( 1 + 6.37T + 47T^{2} \)
53 \( 1 - 8.11T + 53T^{2} \)
59 \( 1 + 6.11T + 59T^{2} \)
61 \( 1 + 0.584T + 61T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + 4.06T + 71T^{2} \)
73 \( 1 - 5.02T + 73T^{2} \)
79 \( 1 - 2.79T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 - 4.54T + 89T^{2} \)
97 \( 1 + 10.5T + 97T^{2} \)
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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

−8.399728719152815338691352813534, −7.37443471649256816626536259787, −6.56840998978841907836650937124, −6.15528571557862653209704056507, −5.31974533503813250499733578652, −5.01287857955519259772243748196, −4.25542292174840323888805194439, −2.92511060621549063459893870250, −1.67586424968311578592368314725, −1.18252481555396403696362805177, 1.18252481555396403696362805177, 1.67586424968311578592368314725, 2.92511060621549063459893870250, 4.25542292174840323888805194439, 5.01287857955519259772243748196, 5.31974533503813250499733578652, 6.15528571557862653209704056507, 6.56840998978841907836650937124, 7.37443471649256816626536259787, 8.399728719152815338691352813534

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