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Properties

Label 4-3850e2-1.1-c1e2-0-17
Degree $4$
Conductor $14822500$
Sign $1$
Analytic cond. $945.095$
Root an. cond. $5.54458$
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  − 4-s + 5·9-s + 2·11-s + 16-s + 20·29-s + 14·31-s − 5·36-s + 4·41-s − 2·44-s − 49-s − 10·59-s + 4·61-s − 64-s + 4·71-s − 10·79-s + 16·81-s + 10·99-s − 6·101-s − 20·116-s + 3·121-s − 14·124-s + 127-s + 131-s + 137-s + 139-s + 5·144-s + 149-s + ⋯
L(s)  = 1  − 1/2·4-s + 5/3·9-s + 0.603·11-s + 1/4·16-s + 3.71·29-s + 2.51·31-s − 5/6·36-s + 0.624·41-s − 0.301·44-s − 1/7·49-s − 1.30·59-s + 0.512·61-s − 1/8·64-s + 0.474·71-s − 1.12·79-s + 16/9·81-s + 1.00·99-s − 0.597·101-s − 1.85·116-s + 3/11·121-s − 1.25·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5/12·144-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14822500\)    =    \(2^{2} \cdot 5^{4} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(945.095\)
Root analytic conductor: \(5.54458\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14822500,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.766387474\)
\(L(\frac12)\) \(\approx\) \(3.766387474\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
7$C_2$ \( 1 + T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.3.a_af
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.13.a_k
17$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.17.a_az
19$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.19.a_bm
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.29.au_gc
31$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.31.ao_eh
37$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.37.a_acn
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.41.ae_di
43$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \) 2.43.a_af
47$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \) 2.47.a_adh
53$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.53.a_az
59$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.59.k_fn
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.61.ae_ew
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \) 2.67.a_acs
71$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \) 2.71.ae_fq
73$C_2^2$ \( 1 - 25 T^{2} + p^{2} T^{4} \) 2.73.a_az
79$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.79.k_hb
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \) 2.83.a_afa
89$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.89.a_gw
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \) 2.97.a_ahi
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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

−8.596473926000624532606022497200, −8.340177375063835868609467998473, −7.943238234818647654080572489025, −7.71280349884212201208567960156, −7.05344707250717195239929398577, −6.72125213113334892972559048624, −6.67750464872572029760257228612, −5.99400642806858103432345363817, −5.98017964958354266558992430066, −4.97051501783891466187671284875, −4.83844141311108012716172212146, −4.42946532007662951490636689804, −4.34355928481966607717319474831, −3.71654280228839823234437195987, −3.25406590173642502167862157051, −2.63161551854811937983130351547, −2.43601804965468171397505533563, −1.34395316703829617433159156303, −1.24947080622664638296193395741, −0.67307536329080186556696887164, 0.67307536329080186556696887164, 1.24947080622664638296193395741, 1.34395316703829617433159156303, 2.43601804965468171397505533563, 2.63161551854811937983130351547, 3.25406590173642502167862157051, 3.71654280228839823234437195987, 4.34355928481966607717319474831, 4.42946532007662951490636689804, 4.83844141311108012716172212146, 4.97051501783891466187671284875, 5.98017964958354266558992430066, 5.99400642806858103432345363817, 6.67750464872572029760257228612, 6.72125213113334892972559048624, 7.05344707250717195239929398577, 7.71280349884212201208567960156, 7.943238234818647654080572489025, 8.340177375063835868609467998473, 8.596473926000624532606022497200

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