Properties

Degree 2
Conductor $ 5 \cdot 11 \cdot 73 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.93·2-s + 3.14·3-s + 1.76·4-s + 5-s + 6.09·6-s − 1.57·7-s − 0.459·8-s + 6.88·9-s + 1.93·10-s + 11-s + 5.54·12-s + 2.21·13-s − 3.04·14-s + 3.14·15-s − 4.41·16-s + 1.29·17-s + 13.3·18-s + 7.65·19-s + 1.76·20-s − 4.94·21-s + 1.93·22-s − 6.25·23-s − 1.44·24-s + 25-s + 4.29·26-s + 12.2·27-s − 2.77·28-s + ⋯
L(s)  = 1  + 1.37·2-s + 1.81·3-s + 0.881·4-s + 0.447·5-s + 2.48·6-s − 0.594·7-s − 0.162·8-s + 2.29·9-s + 0.613·10-s + 0.301·11-s + 1.60·12-s + 0.614·13-s − 0.814·14-s + 0.811·15-s − 1.10·16-s + 0.313·17-s + 3.14·18-s + 1.75·19-s + 0.394·20-s − 1.07·21-s + 0.413·22-s − 1.30·23-s − 0.294·24-s + 0.200·25-s + 0.843·26-s + 2.35·27-s − 0.523·28-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4015\)    =    \(5 \cdot 11 \cdot 73\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{4015} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $8.174015585$
$L(\frac12)$  $\approx$  $8.174015585$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{5,\;11,\;73\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{5,\;11,\;73\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad5 \( 1 - T \)
11 \( 1 - T \)
73 \( 1 - T \)
good2 \( 1 - 1.93T + 2T^{2} \)
3 \( 1 - 3.14T + 3T^{2} \)
7 \( 1 + 1.57T + 7T^{2} \)
13 \( 1 - 2.21T + 13T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
19 \( 1 - 7.65T + 19T^{2} \)
23 \( 1 + 6.25T + 23T^{2} \)
29 \( 1 + 3.27T + 29T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 + 0.134T + 37T^{2} \)
41 \( 1 - 1.69T + 41T^{2} \)
43 \( 1 + 7.90T + 43T^{2} \)
47 \( 1 + 2.16T + 47T^{2} \)
53 \( 1 - 8.22T + 53T^{2} \)
59 \( 1 - 1.15T + 59T^{2} \)
61 \( 1 - 3.61T + 61T^{2} \)
67 \( 1 + 5.50T + 67T^{2} \)
71 \( 1 + 9.92T + 71T^{2} \)
79 \( 1 - 3.22T + 79T^{2} \)
83 \( 1 + 4.72T + 83T^{2} \)
89 \( 1 + 1.80T + 89T^{2} \)
97 \( 1 + 0.996T + 97T^{2} \)
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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

−8.454340267815787250522697574891, −7.69871429974607607817966543208, −6.89387007776346368897062852994, −6.16938457417783421862681071419, −5.36689293399825161104389951838, −4.38171107842967897932469955135, −3.63976130533281130937929015611, −3.19995886975511444021401281964, −2.47252959794111723643472453834, −1.44218758315657243274436995946, 1.44218758315657243274436995946, 2.47252959794111723643472453834, 3.19995886975511444021401281964, 3.63976130533281130937929015611, 4.38171107842967897932469955135, 5.36689293399825161104389951838, 6.16938457417783421862681071419, 6.89387007776346368897062852994, 7.69871429974607607817966543208, 8.454340267815787250522697574891

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