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  + 1.73·2-s − 2.10·3-s + 0.996·4-s + 5-s − 3.64·6-s + 4.00·7-s − 1.73·8-s + 1.43·9-s + 1.73·10-s + 11-s − 2.09·12-s − 4.50·13-s + 6.94·14-s − 2.10·15-s − 4.99·16-s + 2.39·17-s + 2.48·18-s + 1.18·19-s + 0.996·20-s − 8.44·21-s + 1.73·22-s + 4.98·23-s + 3.65·24-s + 25-s − 7.79·26-s + 3.29·27-s + 3.99·28-s + ⋯
L(s)  = 1  + 1.22·2-s − 1.21·3-s + 0.498·4-s + 0.447·5-s − 1.48·6-s + 1.51·7-s − 0.614·8-s + 0.478·9-s + 0.547·10-s + 0.301·11-s − 0.605·12-s − 1.24·13-s + 1.85·14-s − 0.543·15-s − 1.24·16-s + 0.579·17-s + 0.585·18-s + 0.271·19-s + 0.222·20-s − 1.84·21-s + 0.369·22-s + 1.03·23-s + 0.746·24-s + 0.200·25-s − 1.52·26-s + 0.634·27-s + 0.755·28-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  =  $(2,\ 4015,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.666022971$
$L(\frac12)$  $\approx$  $2.666022971$
$L(\frac{3}{2})$   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 \( 1 - 1.73T + 2T^{2} \)
3 \( 1 + 2.10T + 3T^{2} \)
7 \( 1 - 4.00T + 7T^{2} \)
13 \( 1 + 4.50T + 13T^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 - 1.18T + 19T^{2} \)
23 \( 1 - 4.98T + 23T^{2} \)
29 \( 1 + 7.03T + 29T^{2} \)
31 \( 1 - 6.23T + 31T^{2} \)
37 \( 1 - 9.94T + 37T^{2} \)
41 \( 1 - 0.0458T + 41T^{2} \)
43 \( 1 + 6.88T + 43T^{2} \)
47 \( 1 + 1.48T + 47T^{2} \)
53 \( 1 - 4.74T + 53T^{2} \)
59 \( 1 - 6.38T + 59T^{2} \)
61 \( 1 + 2.56T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
79 \( 1 - 9.23T + 79T^{2} \)
83 \( 1 - 9.42T + 83T^{2} \)
89 \( 1 + 7.83T + 89T^{2} \)
97 \( 1 + 2.79T + 97T^{2} \)
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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

−8.325576068024303527853135017373, −7.46012047734970557061702999614, −6.68837289166988666994670295985, −5.87696747112990872030670431633, −5.28498674949643659864546692572, −4.88643406527941717506661904840, −4.31530203696666348017249707800, −3.08119277786315029194809813415, −2.08784976141989273229761696540, −0.843568148032536821777653556235, 0.843568148032536821777653556235, 2.08784976141989273229761696540, 3.08119277786315029194809813415, 4.31530203696666348017249707800, 4.88643406527941717506661904840, 5.28498674949643659864546692572, 5.87696747112990872030670431633, 6.68837289166988666994670295985, 7.46012047734970557061702999614, 8.325576068024303527853135017373

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