Properties

Degree 2
Conductor $ 3 \cdot 13 \cdot 103 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.39·2-s − 3-s − 0.0589·4-s + 0.623·5-s − 1.39·6-s − 3.51·7-s − 2.86·8-s + 9-s + 0.868·10-s − 5.58·11-s + 0.0589·12-s + 13-s − 4.89·14-s − 0.623·15-s − 3.87·16-s − 2.32·17-s + 1.39·18-s + 1.21·19-s − 0.0367·20-s + 3.51·21-s − 7.78·22-s + 3.37·23-s + 2.86·24-s − 4.61·25-s + 1.39·26-s − 27-s + 0.206·28-s + ⋯
L(s)  = 1  + 0.985·2-s − 0.577·3-s − 0.0294·4-s + 0.278·5-s − 0.568·6-s − 1.32·7-s − 1.01·8-s + 0.333·9-s + 0.274·10-s − 1.68·11-s + 0.0170·12-s + 0.277·13-s − 1.30·14-s − 0.160·15-s − 0.969·16-s − 0.564·17-s + 0.328·18-s + 0.278·19-s − 0.00821·20-s + 0.765·21-s − 1.65·22-s + 0.704·23-s + 0.585·24-s − 0.922·25-s + 0.273·26-s − 0.192·27-s + 0.0390·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4017\)    =    \(3 \cdot 13 \cdot 103\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4017} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4017,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.086991042$
$L(\frac12)$  $\approx$  $1.086991042$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{3,\;13,\;103\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{3,\;13,\;103\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad3 \( 1 + T \)
13 \( 1 - T \)
103 \( 1 + T \)
good2 \( 1 - 1.39T + 2T^{2} \)
5 \( 1 - 0.623T + 5T^{2} \)
7 \( 1 + 3.51T + 7T^{2} \)
11 \( 1 + 5.58T + 11T^{2} \)
17 \( 1 + 2.32T + 17T^{2} \)
19 \( 1 - 1.21T + 19T^{2} \)
23 \( 1 - 3.37T + 23T^{2} \)
29 \( 1 - 0.198T + 29T^{2} \)
31 \( 1 - 7.68T + 31T^{2} \)
37 \( 1 + 6.15T + 37T^{2} \)
41 \( 1 + 0.138T + 41T^{2} \)
43 \( 1 - 2.99T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 4.89T + 53T^{2} \)
59 \( 1 + 2.53T + 59T^{2} \)
61 \( 1 + 10.9T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 13.3T + 73T^{2} \)
79 \( 1 + 1.41T + 79T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 - 5.72T + 89T^{2} \)
97 \( 1 + 6.43T + 97T^{2} \)
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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

−8.456088021200898651770413877844, −7.53160814206624815518611021249, −6.59664899983963024270691883577, −6.11692039868298599974252294095, −5.38752588880027726934642264954, −4.85831794121819717325169311847, −3.89885019349365144049826403287, −3.08579082216395025839385753320, −2.38337957040443758732730268798, −0.49786330155549241147428421295, 0.49786330155549241147428421295, 2.38337957040443758732730268798, 3.08579082216395025839385753320, 3.89885019349365144049826403287, 4.85831794121819717325169311847, 5.38752588880027726934642264954, 6.11692039868298599974252294095, 6.59664899983963024270691883577, 7.53160814206624815518611021249, 8.456088021200898651770413877844

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