Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
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 + 4-s + 7-s − 8-s − 4.89·11-s + 0.449·13-s − 14-s + 16-s + 2·17-s + 6.44·19-s + 4.89·22-s − 6.89·23-s − 0.449·26-s + 28-s + 2.89·29-s − 0.898·31-s − 32-s − 2·34-s − 2·37-s − 6.44·38-s + 10.8·41-s − 8.89·43-s − 4.89·44-s + 6.89·46-s − 0.898·47-s + 49-s + 0.449·52-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 0.377·7-s − 0.353·8-s − 1.47·11-s + 0.124·13-s − 0.267·14-s + 0.250·16-s + 0.485·17-s + 1.47·19-s + 1.04·22-s − 1.43·23-s − 0.0881·26-s + 0.188·28-s + 0.538·29-s − 0.161·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s − 1.04·38-s + 1.70·41-s − 1.35·43-s − 0.738·44-s + 1.01·46-s − 0.131·47-s + 0.142·49-s + 0.0623·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.191303430$
$L(\frac12)$  $\approx$  $1.191303430$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 - 0.449T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 6.44T + 19T^{2} \)
23 \( 1 + 6.89T + 23T^{2} \)
29 \( 1 - 2.89T + 29T^{2} \)
31 \( 1 + 0.898T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 + 8.89T + 43T^{2} \)
47 \( 1 + 0.898T + 47T^{2} \)
53 \( 1 + 1.10T + 53T^{2} \)
59 \( 1 - 6.44T + 59T^{2} \)
61 \( 1 - 8.44T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 - 6.89T + 73T^{2} \)
79 \( 1 + 2.89T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 3.79T + 97T^{2} \)
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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

−8.482695314716223891060835874297, −7.954365024723372766470644006249, −7.50278031899873678476055597616, −6.55043219412069728060396235264, −5.55099238941352453213462199316, −5.10518644789293325280282707379, −3.83692107553771915336431592214, −2.86580594149057280301441010398, −1.98663644315735309451399381304, −0.72590807720796895153372359854, 0.72590807720796895153372359854, 1.98663644315735309451399381304, 2.86580594149057280301441010398, 3.83692107553771915336431592214, 5.10518644789293325280282707379, 5.55099238941352453213462199316, 6.55043219412069728060396235264, 7.50278031899873678476055597616, 7.954365024723372766470644006249, 8.482695314716223891060835874297

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