Properties

Label 4-95e2-1.1-c1e2-0-4
Degree $4$
Conductor $9025$
Sign $-1$
Analytic cond. $0.575441$
Root an. cond. $0.870964$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 3·5-s − 9-s − 6·11-s − 19-s + 6·20-s + 4·25-s + 9·29-s − 12·31-s + 2·36-s + 12·44-s + 3·45-s + 2·49-s + 18·55-s + 9·59-s − 19·61-s + 8·64-s + 9·71-s + 2·76-s + 9·79-s − 8·81-s − 9·89-s + 3·95-s + 6·99-s − 8·100-s + 3·101-s − 27·109-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s − 1/3·9-s − 1.80·11-s − 0.229·19-s + 1.34·20-s + 4/5·25-s + 1.67·29-s − 2.15·31-s + 1/3·36-s + 1.80·44-s + 0.447·45-s + 2/7·49-s + 2.42·55-s + 1.17·59-s − 2.43·61-s + 64-s + 1.06·71-s + 0.229·76-s + 1.01·79-s − 8/9·81-s − 0.953·89-s + 0.307·95-s + 0.603·99-s − 4/5·100-s + 0.298·101-s − 2.58·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9025\)    =    \(5^{2} \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(0.575441\)
Root analytic conductor: \(0.870964\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 9025,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad5$C_2$ \( 1 + 3 T + p T^{2} \)
19$C_2$ \( 1 + T + p T^{2} \)
good2$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \) 2.2.a_c
3$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.3.a_b
7$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.7.a_ac
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \) 2.11.g_bf
13$C_2^2$ \( 1 + 23 T^{2} + p^{2} T^{4} \) 2.13.a_x
17$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \) 2.17.a_b
23$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \) 2.23.a_ar
29$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) 2.29.aj_cy
31$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.31.m_dt
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.37.a_ak
41$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.41.a_cv
43$C_2^2$ \( 1 + 7 T^{2} + p^{2} T^{4} \) 2.43.a_h
47$C_2^2$ \( 1 + 79 T^{2} + p^{2} T^{4} \) 2.47.a_db
53$C_2^2$ \( 1 - 73 T^{2} + p^{2} T^{4} \) 2.53.a_acv
59$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.59.aj_de
61$C_2$$\times$$C_2$ \( ( 1 + 5 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.61.t_hk
67$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.67.a_abf
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \) 2.71.aj_fm
73$C_2^2$ \( 1 - 35 T^{2} + p^{2} T^{4} \) 2.73.a_abj
79$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.aj_gw
83$C_2^2$ \( 1 + 142 T^{2} + p^{2} T^{4} \) 2.83.a_fm
89$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.j_q
97$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.97.a_an
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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

−11.20596463141761880232308913941, −10.86663596298499142634886865184, −10.32692888075682554881955096544, −9.600825932349098521069904813407, −8.902747827759237921715267272211, −8.434380274846306746371413761130, −7.907590442336435129393185114445, −7.46730325217072203305692834103, −6.66583993396032814586673339969, −5.57733380179038841837323712732, −5.03077182024000936944442191827, −4.36902019806103263721464598563, −3.60401610781285747484879091023, −2.63792299456794529108614129708, 0, 2.63792299456794529108614129708, 3.60401610781285747484879091023, 4.36902019806103263721464598563, 5.03077182024000936944442191827, 5.57733380179038841837323712732, 6.66583993396032814586673339969, 7.46730325217072203305692834103, 7.907590442336435129393185114445, 8.434380274846306746371413761130, 8.902747827759237921715267272211, 9.600825932349098521069904813407, 10.32692888075682554881955096544, 10.86663596298499142634886865184, 11.20596463141761880232308913941

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