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Properties

Label 4-3328e2-1.1-c1e2-0-17
Degree $4$
Conductor $11075584$
Sign $1$
Analytic cond. $706.188$
Root an. cond. $5.15501$
Motivic weight $1$
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  + 4·5-s − 6·9-s + 2·13-s + 8·17-s + 2·25-s + 4·29-s + 12·37-s + 4·41-s − 24·45-s + 4·49-s + 16·53-s + 8·65-s + 12·73-s + 27·81-s + 32·85-s + 20·89-s + 36·97-s + 8·101-s − 4·109-s + 32·113-s − 12·117-s − 4·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.78·5-s − 2·9-s + 0.554·13-s + 1.94·17-s + 2/5·25-s + 0.742·29-s + 1.97·37-s + 0.624·41-s − 3.57·45-s + 4/7·49-s + 2.19·53-s + 0.992·65-s + 1.40·73-s + 3·81-s + 3.47·85-s + 2.11·89-s + 3.65·97-s + 0.796·101-s − 0.383·109-s + 3.01·113-s − 1.10·117-s − 0.363·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11075584\)    =    \(2^{16} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(706.188\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11075584,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.434656629\)
\(L(\frac12)\) \(\approx\) \(4.434656629\)
\(L(\frac{3}{2})\) 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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
13$C_1$ \( ( 1 - T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.5.ae_o
7$C_2^2$ \( 1 - 4 T^{2} + p^{2} T^{4} \) 2.7.a_ae
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.11.a_e
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.19.a_u
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.29.ae_ck
31$C_2^2$ \( 1 + 44 T^{2} + p^{2} T^{4} \) 2.31.a_bs
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.37.am_eg
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \) 2.43.a_o
47$C_2^2$ \( 1 - 68 T^{2} + p^{2} T^{4} \) 2.47.a_acq
53$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.53.aq_go
59$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.59.a_dw
61$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.61.a_es
67$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \) 2.67.a_em
71$C_2^2$ \( 1 + 124 T^{2} + p^{2} T^{4} \) 2.71.a_eu
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.73.am_ha
79$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \) 2.79.a_di
83$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.83.a_e
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.89.au_ks
97$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \) 2.97.abk_ty
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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

−8.880065425571858706942799890627, −8.461737782048740137861198958979, −8.031940714237094533143137271694, −7.906473759734572903941838466636, −7.35072586761150164485572773113, −6.92523747964179791876431093472, −6.16410290635155308074101184602, −6.01119277230305005143155908568, −5.94246221186174764485190134229, −5.67047705650889064481363951379, −4.96840338041671652676169804616, −4.96706583586978258007546045841, −4.09430599752507582068969888485, −3.46556431099378018558028922610, −3.33491292055218823431806026057, −2.66577979053547312977368486640, −2.21171521375535795957682057847, −2.07705917948157762578129462099, −0.954985710560402240773365680671, −0.806189440881641009930242873914, 0.806189440881641009930242873914, 0.954985710560402240773365680671, 2.07705917948157762578129462099, 2.21171521375535795957682057847, 2.66577979053547312977368486640, 3.33491292055218823431806026057, 3.46556431099378018558028922610, 4.09430599752507582068969888485, 4.96706583586978258007546045841, 4.96840338041671652676169804616, 5.67047705650889064481363951379, 5.94246221186174764485190134229, 6.01119277230305005143155908568, 6.16410290635155308074101184602, 6.92523747964179791876431093472, 7.35072586761150164485572773113, 7.906473759734572903941838466636, 8.031940714237094533143137271694, 8.461737782048740137861198958979, 8.880065425571858706942799890627

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