Properties

Label 4-770e2-1.1-c1e2-0-4
Degree $4$
Conductor $592900$
Sign $1$
Analytic cond. $37.8038$
Root an. cond. $2.47961$
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-s − 3-s + 5-s + 6-s + 5·7-s + 8-s + 3·9-s − 10-s − 11-s − 2·13-s − 5·14-s − 15-s − 16-s − 3·18-s − 2·19-s − 5·21-s + 22-s + 6·23-s − 24-s + 2·26-s − 8·27-s − 18·29-s + 30-s + 4·31-s + 33-s + 5·35-s + 4·37-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 9-s − 0.316·10-s − 0.301·11-s − 0.554·13-s − 1.33·14-s − 0.258·15-s − 1/4·16-s − 0.707·18-s − 0.458·19-s − 1.09·21-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 0.392·26-s − 1.53·27-s − 3.34·29-s + 0.182·30-s + 0.718·31-s + 0.174·33-s + 0.845·35-s + 0.657·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(592900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(37.8038\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{770} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 592900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.387605489\)
\(L(\frac12)\) \(\approx\) \(1.387605489\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_2$ \( 1 - 5 T + p T^{2} \)
11$C_2$ \( 1 + T + T^{2} \)
good3$C_2^2$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2^2$ \( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
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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

−10.61449610944512178646940220150, −10.07796453091352308826375609508, −9.679683399032555341836053891848, −9.139125322022559606312743260315, −9.090802354248279677439902182566, −8.367684695295255592829198576795, −7.83274350919790756672590506936, −7.51789791286739638763228651404, −7.37561853487199537228550294581, −6.79140284589459395358371390830, −5.84137456448527130638979387294, −5.81734617411844556639530652676, −4.97021534511446601711356865089, −4.94751381703905328827684690702, −4.12497470268647556198418499769, −3.91109391330071955111102378890, −2.68931844305498549818898702149, −1.96923828456132310031056932113, −1.64494880361374577318458564940, −0.73395867792172558887207203549, 0.73395867792172558887207203549, 1.64494880361374577318458564940, 1.96923828456132310031056932113, 2.68931844305498549818898702149, 3.91109391330071955111102378890, 4.12497470268647556198418499769, 4.94751381703905328827684690702, 4.97021534511446601711356865089, 5.81734617411844556639530652676, 5.84137456448527130638979387294, 6.79140284589459395358371390830, 7.37561853487199537228550294581, 7.51789791286739638763228651404, 7.83274350919790756672590506936, 8.367684695295255592829198576795, 9.090802354248279677439902182566, 9.139125322022559606312743260315, 9.679683399032555341836053891848, 10.07796453091352308826375609508, 10.61449610944512178646940220150

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