Properties

Label 4-145152-1.1-c1e2-0-14
Degree $4$
Conductor $145152$
Sign $1$
Analytic cond. $9.25501$
Root an. cond. $1.74419$
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  + 5·7-s + 12·19-s + 2·25-s − 8·31-s − 12·37-s + 12·43-s + 14·49-s − 12·61-s + 8·67-s − 8·79-s − 8·103-s + 4·109-s + 2·121-s + ⋯
L(s)  = 1  + 1.88·7-s + 2.75·19-s + 2/5·25-s − 1.43·31-s − 1.97·37-s + 1.82·43-s + 2·49-s − 1.53·61-s + 0.977·67-s − 0.900·79-s − 0.788·103-s + 0.383·109-s + 2/11·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 145152 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145152 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(145152\)    =    \(2^{8} \cdot 3^{4} \cdot 7\)
Sign: $1$
Analytic conductor: \(9.25501\)
Root analytic conductor: \(1.74419\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 145152,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.188510757\)
\(L(\frac12)\) \(\approx\) \(2.188510757\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
7$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 4 T + p T^{2} ) \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
11$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.11.a_ac
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.13.a_w
17$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.17.a_ak
19$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.19.am_cs
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.23.a_ac
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.29.a_ag
31$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.31.i_ck
37$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.m_dq
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.a_as
43$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) 2.43.am_di
47$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.47.a_bu
53$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.53.a_abu
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.59.a_ak
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.m_fm
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.71.a_abi
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.73.a_fm
79$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.79.i_gc
83$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.83.a_g
89$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.89.a_bu
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.97.a_hi
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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

−9.134803102065345052447420465800, −8.904420087550581034572732623942, −8.337806533319699121595152839388, −7.71174954182317866915094856428, −7.39330514227390535766462482348, −7.22159161985493072751553303991, −6.27999109778613833497992618178, −5.50026338013035836046779699541, −5.27220853215504505998094578272, −4.88853847412390723429004693687, −4.09919993919492656198401600205, −3.48693318706905841377286487654, −2.73631288380822660798226588670, −1.76777294938515484010129385102, −1.16910645003912789161115085105, 1.16910645003912789161115085105, 1.76777294938515484010129385102, 2.73631288380822660798226588670, 3.48693318706905841377286487654, 4.09919993919492656198401600205, 4.88853847412390723429004693687, 5.27220853215504505998094578272, 5.50026338013035836046779699541, 6.27999109778613833497992618178, 7.22159161985493072751553303991, 7.39330514227390535766462482348, 7.71174954182317866915094856428, 8.337806533319699121595152839388, 8.904420087550581034572732623942, 9.134803102065345052447420465800

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