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Properties

Label 4-1812608-1.1-c1e2-0-11
Degree $4$
Conductor $1812608$
Sign $1$
Analytic cond. $115.573$
Root an. cond. $3.27879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 8-s − 6·9-s − 4·11-s + 16-s − 2·17-s + 6·18-s − 4·19-s + 4·22-s − 6·25-s − 32-s + 2·34-s − 6·36-s + 4·38-s − 12·41-s + 8·43-s − 4·44-s + 49-s + 6·50-s + 20·59-s + 64-s + 16·67-s − 2·68-s + 6·72-s − 20·73-s − 4·76-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.353·8-s − 2·9-s − 1.20·11-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.917·19-s + 0.852·22-s − 6/5·25-s − 0.176·32-s + 0.342·34-s − 36-s + 0.648·38-s − 1.87·41-s + 1.21·43-s − 0.603·44-s + 1/7·49-s + 0.848·50-s + 2.60·59-s + 1/8·64-s + 1.95·67-s − 0.242·68-s + 0.707·72-s − 2.34·73-s − 0.458·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1812608 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1812608\)    =    \(2^{7} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(115.573\)
Root analytic conductor: \(3.27879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 1812608,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
17$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.a_g
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.11.e_ba
13$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.13.a_ba
19$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.19.e_bq
23$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.23.a_as
29$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.29.a_cg
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.a_ac
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.41.m_eo
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.43.ai_dy
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.47.a_be
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.53.a_cs
59$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.59.au_ik
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.a_w
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \) 2.67.aq_hq
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.71.a_ew
73$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.73.u_jm
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.a_fm
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.83.m_hu
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \) 2.89.m_ig
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \) 2.97.bc_pa
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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

−7.51606420165582292370699563489, −6.80584658211117866681394742191, −6.76626773482443698632812811275, −6.02792996427685742234709543306, −5.62672058490049396710575997042, −5.39648301888641876936794255128, −4.97407899427248836000420043781, −3.96511449610299471402331111472, −3.90223646589081978007515090504, −2.91658480070591982635889937604, −2.56493236008968775596152231873, −2.32487504750581987201871119964, −1.38611477851746612785212114986, 0, 0, 1.38611477851746612785212114986, 2.32487504750581987201871119964, 2.56493236008968775596152231873, 2.91658480070591982635889937604, 3.90223646589081978007515090504, 3.96511449610299471402331111472, 4.97407899427248836000420043781, 5.39648301888641876936794255128, 5.62672058490049396710575997042, 6.02792996427685742234709543306, 6.76626773482443698632812811275, 6.80584658211117866681394742191, 7.51606420165582292370699563489

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