Properties

Label 4-35e2-1.1-c4e2-0-0
Degree $4$
Conductor $1225$
Sign $1$
Analytic cond. $13.0895$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 10·3-s + 26·4-s − 10·5-s − 70·7-s − 87·9-s + 178·11-s − 260·12-s + 10·13-s + 100·15-s + 420·16-s + 970·17-s − 260·20-s + 700·21-s − 525·25-s + 1.93e3·27-s − 1.82e3·28-s + 382·29-s − 1.78e3·33-s + 700·35-s − 2.26e3·36-s − 100·39-s + 4.62e3·44-s + 870·45-s + 4.39e3·47-s − 4.20e3·48-s + 2.49e3·49-s − 9.70e3·51-s + ⋯
L(s)  = 1  − 1.11·3-s + 13/8·4-s − 2/5·5-s − 1.42·7-s − 1.07·9-s + 1.47·11-s − 1.80·12-s + 0.0591·13-s + 4/9·15-s + 1.64·16-s + 3.35·17-s − 0.649·20-s + 1.58·21-s − 0.839·25-s + 2.64·27-s − 2.32·28-s + 0.454·29-s − 1.63·33-s + 4/7·35-s − 1.74·36-s − 0.0657·39-s + 2.39·44-s + 0.429·45-s + 1.98·47-s − 1.82·48-s + 1.04·49-s − 3.72·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(13.0895\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1225,\ (\ :2, 2),\ 1)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.395016627\)
\(L(\frac12)\) \(\approx\) \(1.395016627\)
\(L(3)\) 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)$
bad5$C_2$ \( 1 + 2 p T + p^{4} T^{2} \)
7$C_2$ \( 1 + 10 p T + p^{4} T^{2} \)
good2$C_2^2$ \( 1 - 13 p T^{2} + p^{8} T^{4} \)
3$C_2$ \( ( 1 + 5 T + p^{4} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 89 T + p^{4} T^{2} )^{2} \)
13$C_2$ \( ( 1 - 5 T + p^{4} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 485 T + p^{4} T^{2} )^{2} \)
19$C_2^2$ \( 1 - 212042 T^{2} + p^{8} T^{4} \)
23$C_2^2$ \( 1 - 68906 T^{2} + p^{8} T^{4} \)
29$C_2$ \( ( 1 - 191 T + p^{4} T^{2} )^{2} \)
31$C_2^2$ \( 1 - 737642 T^{2} + p^{8} T^{4} \)
37$C_2^2$ \( 1 - 794 p^{2} T^{2} + p^{8} T^{4} \)
41$C_2^2$ \( 1 + 2845078 T^{2} + p^{8} T^{4} \)
43$C_2^2$ \( 1 - 6695306 T^{2} + p^{8} T^{4} \)
47$C_2$ \( ( 1 - 2195 T + p^{4} T^{2} )^{2} \)
53$C_2^2$ \( 1 - 13261538 T^{2} + p^{8} T^{4} \)
59$C_2^2$ \( 1 - 11092322 T^{2} + p^{8} T^{4} \)
61$C_2^2$ \( 1 - 23947082 T^{2} + p^{8} T^{4} \)
67$C_2^2$ \( 1 - 36108866 T^{2} + p^{8} T^{4} \)
71$C_2$ \( ( 1 - 4454 T + p^{4} T^{2} )^{2} \)
73$C_2$ \( ( 1 + 8650 T + p^{4} T^{2} )^{2} \)
79$C_2$ \( ( 1 - 5561 T + p^{4} T^{2} )^{2} \)
83$C_2$ \( ( 1 + 1990 T + p^{4} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 124831082 T^{2} + p^{8} T^{4} \)
97$C_2$ \( ( 1 + 9235 T + p^{4} T^{2} )^{2} \)
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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

−16.32059356744224888236700699846, −15.75967767595344737108854696118, −14.98014542926527280332458792812, −14.34350143146305992649782016101, −13.95439614707832904826478242169, −12.48867412441494126891130975416, −12.18780273573968715198874847638, −11.82903793454783615842268947815, −11.44070189882575411020251376976, −10.57242669315692193649474573994, −10.07097693322161259283997131319, −9.242173355310444402337589436805, −8.134074552846417593436736188004, −7.30324564692594695078584080633, −6.57354826322717129890171805563, −5.84053442731215097730335420704, −5.70706770608959593601195736919, −3.57956700653466011907015963814, −2.94758631634964076960185359270, −0.928285038264232944680980413603, 0.928285038264232944680980413603, 2.94758631634964076960185359270, 3.57956700653466011907015963814, 5.70706770608959593601195736919, 5.84053442731215097730335420704, 6.57354826322717129890171805563, 7.30324564692594695078584080633, 8.134074552846417593436736188004, 9.242173355310444402337589436805, 10.07097693322161259283997131319, 10.57242669315692193649474573994, 11.44070189882575411020251376976, 11.82903793454783615842268947815, 12.18780273573968715198874847638, 12.48867412441494126891130975416, 13.95439614707832904826478242169, 14.34350143146305992649782016101, 14.98014542926527280332458792812, 15.75967767595344737108854696118, 16.32059356744224888236700699846

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