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Properties

Label 4-1813185-1.1-c1e2-0-7
Degree $4$
Conductor $1813185$
Sign $1$
Analytic cond. $115.610$
Root an. cond. $3.27905$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·4-s + 3·5-s − 4·11-s + 5·16-s + 9·20-s + 2·25-s + 12·31-s + 5·37-s − 12·44-s + 4·47-s − 10·49-s − 8·53-s − 12·55-s − 4·59-s + 3·64-s + 20·67-s + 24·71-s + 15·80-s − 8·97-s + 6·100-s + 16·103-s + 8·113-s + 5·121-s + 36·124-s − 10·125-s + 127-s + 131-s + ⋯
L(s)  = 1  + 3/2·4-s + 1.34·5-s − 1.20·11-s + 5/4·16-s + 2.01·20-s + 2/5·25-s + 2.15·31-s + 0.821·37-s − 1.80·44-s + 0.583·47-s − 1.42·49-s − 1.09·53-s − 1.61·55-s − 0.520·59-s + 3/8·64-s + 2.44·67-s + 2.84·71-s + 1.67·80-s − 0.812·97-s + 3/5·100-s + 1.57·103-s + 0.752·113-s + 5/11·121-s + 3.23·124-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1813185 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1813185 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1813185\)    =    \(3^{4} \cdot 5 \cdot 11^{2} \cdot 37\)
Sign: $1$
Analytic conductor: \(115.610\)
Root analytic conductor: \(3.27905\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1813185,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.356073526\)
\(L(\frac12)\) \(\approx\) \(4.356073526\)
\(L(\frac{3}{2})\) 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 $\mathbf{F}_p$
bad3 \( 1 \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 2 T + p T^{2} ) \)
11$C_2$ \( 1 + 4 T + p T^{2} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 6 T + p T^{2} ) \)
good2$C_2^2$ \( 1 - 3 T^{2} + p^{2} T^{4} \) 2.2.a_ad
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.7.a_k
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.17.a_bi
19$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.19.a_ag
23$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.23.a_bu
29$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.29.a_abe
31$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.31.am_dq
41$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \) 2.41.a_aby
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.43.a_aw
47$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) 2.47.ae_dq
53$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.i_di
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.e_di
61$C_2^2$ \( 1 + 102 T^{2} + p^{2} T^{4} \) 2.61.a_dy
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.67.au_iw
71$C_2$$\times$$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) 2.71.ay_kk
73$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \) 2.73.a_w
79$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \) 2.79.a_bi
83$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.83.a_bu
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.89.a_fm
97$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.97.i_o
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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

−7.84568489441652431477913982813, −7.37482007714619421272336093009, −6.77251905944614922252048269579, −6.44112777480872756524433747349, −6.24326212998732298129468215013, −5.80288176596204430688232059224, −5.20103986226906202863906999266, −5.00062951819933039203240248915, −4.34731118485343529059781561559, −3.54750812282005889748272398621, −2.98602060106649751232150105850, −2.51233627894296348543100554094, −2.23132019577560214363254365875, −1.66903809515371837713892759805, −0.839286366343457752065081402962, 0.839286366343457752065081402962, 1.66903809515371837713892759805, 2.23132019577560214363254365875, 2.51233627894296348543100554094, 2.98602060106649751232150105850, 3.54750812282005889748272398621, 4.34731118485343529059781561559, 5.00062951819933039203240248915, 5.20103986226906202863906999266, 5.80288176596204430688232059224, 6.24326212998732298129468215013, 6.44112777480872756524433747349, 6.77251905944614922252048269579, 7.37482007714619421272336093009, 7.84568489441652431477913982813

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