Properties

Degree $4$
Conductor $275625$
Sign $1$
Motivic weight $3$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·4-s − 9·9-s − 72·11-s − 15·16-s + 248·19-s − 204·29-s − 320·31-s − 63·36-s − 636·41-s − 504·44-s − 49·49-s + 264·59-s + 796·61-s − 553·64-s − 1.44e3·71-s + 1.73e3·76-s + 2.04e3·79-s + 81·81-s − 708·89-s + 648·99-s + 828·101-s − 2.95e3·109-s − 1.42e3·116-s + 1.22e3·121-s − 2.24e3·124-s + 127-s + 131-s + ⋯
L(s)  = 1  + 7/8·4-s − 1/3·9-s − 1.97·11-s − 0.234·16-s + 2.99·19-s − 1.30·29-s − 1.85·31-s − 0.291·36-s − 2.42·41-s − 1.72·44-s − 1/7·49-s + 0.582·59-s + 1.67·61-s − 1.08·64-s − 2.40·71-s + 2.62·76-s + 2.91·79-s + 1/9·81-s − 0.843·89-s + 0.657·99-s + 0.815·101-s − 2.59·109-s − 1.14·116-s + 0.921·121-s − 1.62·124-s + 0.000698·127-s + 0.000666·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(275625\)    =    \(3^{2} \cdot 5^{4} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(3\)
Character: induced by $\chi_{525} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 275625,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.438440857\)
\(L(\frac12)\) \(\approx\) \(1.438440857\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad3$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + p^{2} T^{2} \)
good2$C_2^2$ \( 1 - 7 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 36 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 3238 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 8062 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 124 T + p^{3} T^{2} )^{2} \)
23$C_2$ \( ( 1 - p^{3} T^{2} )^{2} \)
29$C_2$ \( ( 1 + 102 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 160 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 57098 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 318 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 87190 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 150046 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 49750 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 132 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 398 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 593062 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 720 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 526030 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1024 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1101958 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 354 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1743550 T^{2} + p^{6} T^{4} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.66193279970819223325488061387, −10.32257245525055849242410444051, −9.865879719944274928991822775243, −9.251331029544100009051416102202, −9.143439904311811354957570134750, −8.090003120939171846755625404253, −8.079302956043925042421468521326, −7.45558017372683787280519828667, −6.99400254499154043103699204717, −6.91869649410433302313298807450, −5.78029321645497964324701007115, −5.56199206591429851803549688016, −5.23618932718648716172114894495, −4.73330100481057940440338863548, −3.55039658846958212328398534473, −3.37123498533767392642043421118, −2.69734339914058008430854870133, −2.11091259680074434324615154753, −1.45223355996482604762642215167, −0.34937687866765288249035281407, 0.34937687866765288249035281407, 1.45223355996482604762642215167, 2.11091259680074434324615154753, 2.69734339914058008430854870133, 3.37123498533767392642043421118, 3.55039658846958212328398534473, 4.73330100481057940440338863548, 5.23618932718648716172114894495, 5.56199206591429851803549688016, 5.78029321645497964324701007115, 6.91869649410433302313298807450, 6.99400254499154043103699204717, 7.45558017372683787280519828667, 8.079302956043925042421468521326, 8.090003120939171846755625404253, 9.143439904311811354957570134750, 9.251331029544100009051416102202, 9.865879719944274928991822775243, 10.32257245525055849242410444051, 10.66193279970819223325488061387

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