Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
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 + 3-s + 4-s − 6-s − 8-s + 9-s + 0.585·11-s + 12-s + 3·13-s + 16-s + 0.171·17-s − 18-s − 1.41·19-s − 0.585·22-s + 2.41·23-s − 24-s − 3·26-s + 27-s + 29-s + 0.414·31-s − 32-s + 0.585·33-s − 0.171·34-s + 36-s − 7.07·37-s + 1.41·38-s + 3·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 0.176·11-s + 0.288·12-s + 0.832·13-s + 0.250·16-s + 0.0416·17-s − 0.235·18-s − 0.324·19-s − 0.124·22-s + 0.503·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.185·29-s + 0.0743·31-s − 0.176·32-s + 0.101·33-s − 0.0294·34-s + 0.166·36-s − 1.16·37-s + 0.229·38-s + 0.480·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(1.949140034\)
\(L(\frac12)\)  \(\approx\)  \(1.949140034\)
\(L(\frac{3}{2})\)   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 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 0.585T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 0.171T + 17T^{2} \)
19 \( 1 + 1.41T + 19T^{2} \)
23 \( 1 - 2.41T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 - 0.414T + 31T^{2} \)
37 \( 1 + 7.07T + 37T^{2} \)
41 \( 1 + 0.171T + 41T^{2} \)
43 \( 1 - 4.41T + 43T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 - 1.34T + 53T^{2} \)
59 \( 1 + 1.24T + 59T^{2} \)
61 \( 1 + 5.82T + 61T^{2} \)
67 \( 1 - 7.31T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 0.928T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.041555851186913625288322312294, −7.31193330571970690133007942501, −6.69733594249245796766209872567, −5.99100082271201054162744445338, −5.14136987488770864214114403276, −4.13248206154617541701432238752, −3.45478792375148122724305579183, −2.59448175784176780295541620585, −1.73016443050541720897329165760, −0.78379502172290067623818240660, 0.78379502172290067623818240660, 1.73016443050541720897329165760, 2.59448175784176780295541620585, 3.45478792375148122724305579183, 4.13248206154617541701432238752, 5.14136987488770864214114403276, 5.99100082271201054162744445338, 6.69733594249245796766209872567, 7.31193330571970690133007942501, 8.041555851186913625288322312294

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