Properties

Label 4-975e2-1.1-c1e2-0-11
Degree $4$
Conductor $950625$
Sign $1$
Analytic cond. $60.6126$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4-s + 4·7-s + 8·8-s − 9-s − 4·13-s − 8·14-s − 7·16-s + 2·18-s + 8·26-s − 4·28-s − 4·29-s − 14·32-s + 36-s + 16·37-s + 8·47-s − 2·49-s + 4·52-s + 32·56-s + 8·58-s + 20·61-s − 4·63-s + 35·64-s + 12·67-s − 8·72-s + 32·73-s − 32·74-s + ⋯
L(s)  = 1  − 1.41·2-s − 1/2·4-s + 1.51·7-s + 2.82·8-s − 1/3·9-s − 1.10·13-s − 2.13·14-s − 7/4·16-s + 0.471·18-s + 1.56·26-s − 0.755·28-s − 0.742·29-s − 2.47·32-s + 1/6·36-s + 2.63·37-s + 1.16·47-s − 2/7·49-s + 0.554·52-s + 4.27·56-s + 1.05·58-s + 2.56·61-s − 0.503·63-s + 35/8·64-s + 1.46·67-s − 0.942·72-s + 3.74·73-s − 3.71·74-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(950625\)    =    \(3^{2} \cdot 5^{4} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(60.6126\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 950625,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7538377455\)
\(L(\frac12)\) \(\approx\) \(0.7538377455\)
\(L(\frac{3}{2})\) 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)$
bad3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
13$C_2$ \( 1 + 4 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
71$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
97$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.849276096279669223049611554917, −9.772790733062023556882501648677, −9.276602864790391214094286383910, −9.215105814924127657895284213617, −8.361976982683085835014285171997, −8.171682684316208674495123942692, −7.944445087031193684887677758044, −7.69105310625206350506702729568, −7.04382829449703029780450372581, −6.66351444443766357764865815522, −5.77461921193379791353830729147, −5.28692197655671010544400843998, −4.82898024210049128183583798973, −4.77854757646123139099314181611, −3.89584692282209882077453422377, −3.74521107810004015453620875232, −2.30102597585143219330979714378, −2.21064982689969629017629978612, −1.14860679041629530404679759551, −0.62795496767986474957423743272, 0.62795496767986474957423743272, 1.14860679041629530404679759551, 2.21064982689969629017629978612, 2.30102597585143219330979714378, 3.74521107810004015453620875232, 3.89584692282209882077453422377, 4.77854757646123139099314181611, 4.82898024210049128183583798973, 5.28692197655671010544400843998, 5.77461921193379791353830729147, 6.66351444443766357764865815522, 7.04382829449703029780450372581, 7.69105310625206350506702729568, 7.944445087031193684887677758044, 8.171682684316208674495123942692, 8.361976982683085835014285171997, 9.215105814924127657895284213617, 9.276602864790391214094286383910, 9.772790733062023556882501648677, 9.849276096279669223049611554917

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