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Properties

Label 4-952e2-1.1-c1e2-0-1
Degree $4$
Conductor $906304$
Sign $1$
Analytic cond. $57.7867$
Root an. cond. $2.75712$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·9-s − 4·13-s − 2·17-s − 8·19-s + 6·25-s − 8·43-s − 24·47-s − 49-s + 28·53-s − 24·59-s + 8·67-s − 5·81-s + 24·83-s + 12·89-s + 12·101-s − 16·103-s − 8·117-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + ⋯
L(s)  = 1  + 2/3·9-s − 1.10·13-s − 0.485·17-s − 1.83·19-s + 6/5·25-s − 1.21·43-s − 3.50·47-s − 1/7·49-s + 3.84·53-s − 3.12·59-s + 0.977·67-s − 5/9·81-s + 2.63·83-s + 1.27·89-s + 1.19·101-s − 1.57·103-s − 0.739·117-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 906304 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 906304 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(906304\)    =    \(2^{6} \cdot 7^{2} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(57.7867\)
Root analytic conductor: \(2.75712\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 906304,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.396893549\)
\(L(\frac12)\) \(\approx\) \(1.396893549\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
17$C_2$ \( 1 + 2 T + p T^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.3.a_ac
5$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.5.a_ag
11$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \) 2.11.a_as
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \) 2.13.e_be
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.19.i_cc
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \) 2.29.a_acc
31$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.31.a_c
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.37.a_acs
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.43.i_dy
47$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.47.y_je
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \) 2.53.abc_lq
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.59.y_kc
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.61.a_aw
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.67.ai_fu
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.73.a_afa
79$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \) 2.79.a_afm
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \) 2.83.ay_ly
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.89.am_ig
97$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.97.a_ahm
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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

−10.42909621063052584220467834502, −9.787631055038512852986802844205, −9.475263683827022844510275621459, −9.007576116643374999327295813758, −8.575609929013050676229577877796, −8.141415524425596874377348804553, −7.85652221302629693662324340989, −7.03136186665212710357962364809, −6.99373181577466169510285717532, −6.43524554507605748585838443025, −6.16348903674374003161698669100, −5.27461664387749374580879984781, −4.96523717164673058973588718804, −4.50550046238098896125693093128, −4.15599797286532655608671227494, −3.36840597926814609027920291707, −2.91369995797359774831254180866, −2.05016709245556178431596412635, −1.80898784762226579827577096394, −0.53751099765746333308399639573, 0.53751099765746333308399639573, 1.80898784762226579827577096394, 2.05016709245556178431596412635, 2.91369995797359774831254180866, 3.36840597926814609027920291707, 4.15599797286532655608671227494, 4.50550046238098896125693093128, 4.96523717164673058973588718804, 5.27461664387749374580879984781, 6.16348903674374003161698669100, 6.43524554507605748585838443025, 6.99373181577466169510285717532, 7.03136186665212710357962364809, 7.85652221302629693662324340989, 8.141415524425596874377348804553, 8.575609929013050676229577877796, 9.007576116643374999327295813758, 9.475263683827022844510275621459, 9.787631055038512852986802844205, 10.42909621063052584220467834502

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