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Properties

Label 4-185e2-1.1-c1e2-0-3
Degree $4$
Conductor $34225$
Sign $1$
Analytic cond. $2.18221$
Root an. cond. $1.21541$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 4·3-s − 4-s − 2·5-s + 8·6-s + 8·8-s + 8·9-s + 4·10-s + 4·12-s − 8·13-s + 8·15-s − 7·16-s − 16·18-s + 2·20-s + 8·23-s − 32·24-s − 25-s + 16·26-s − 12·27-s − 2·29-s − 16·30-s − 12·31-s − 14·32-s − 8·36-s − 2·37-s + 32·39-s − 16·40-s + ⋯
L(s)  = 1  − 1.41·2-s − 2.30·3-s − 1/2·4-s − 0.894·5-s + 3.26·6-s + 2.82·8-s + 8/3·9-s + 1.26·10-s + 1.15·12-s − 2.21·13-s + 2.06·15-s − 7/4·16-s − 3.77·18-s + 0.447·20-s + 1.66·23-s − 6.53·24-s − 1/5·25-s + 3.13·26-s − 2.30·27-s − 0.371·29-s − 2.92·30-s − 2.15·31-s − 2.47·32-s − 4/3·36-s − 0.328·37-s + 5.12·39-s − 2.52·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(34225\)    =    \(5^{2} \cdot 37^{2}\)
Sign: $1$
Analytic conductor: \(2.18221\)
Root analytic conductor: \(1.21541\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 34225,\ (\ :1/2, 1/2),\ 1)\)

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$C_2$ \( 1 + 2 T + p T^{2} \)
37$C_2$ \( 1 + 2 T + p T^{2} \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.2.c_f
3$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.3.e_i
7$C_2^2$ \( 1 + p^{2} T^{4} \) 2.7.a_a
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.11.a_ag
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.13.i_bq
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.17.a_abe
19$C_2^2$ \( 1 + p^{2} T^{4} \) 2.19.a_a
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.23.ai_ck
29$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.29.c_c
31$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.31.m_cu
41$C_2$ \( ( 1 - p T^{2} )^{2} \) 2.41.a_ade
43$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \) 2.43.y_iw
47$C_2^2$ \( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} \) 2.47.aq_ey
53$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) 2.53.s_gg
59$C_2^2$ \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) 2.59.ai_bg
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.61.ac_c
67$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.67.m_cu
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \) 2.71.i_gc
73$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) 2.73.w_ji
79$C_2^2$ \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) 2.79.m_cu
83$C_2^2$ \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) 2.83.e_i
89$C_2^2$ \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) 2.89.c_c
97$C_2^2$ \( 1 - 178 T^{2} + p^{2} T^{4} \) 2.97.a_agw
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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

−12.16863657551990597683003049864, −11.79925645971910819215343496929, −11.05233437961054836011496367652, −11.04441630478008108341163433690, −10.36069300317884076015181233277, −9.842374305639781560858925177247, −9.620012553799449594182753328326, −8.867023978556716664989436402135, −8.459038134695047937455211668892, −7.53375307058247587357249366422, −7.16788345206229107744699076591, −7.14977691777917398809994079094, −5.80487837787603637191963573195, −5.33469939284089197070161284126, −4.69954442347950675612313087224, −4.62713613916430467161249897649, −3.51600518630916943386036690955, −1.56723760771432894279445474181, 0, 0, 1.56723760771432894279445474181, 3.51600518630916943386036690955, 4.62713613916430467161249897649, 4.69954442347950675612313087224, 5.33469939284089197070161284126, 5.80487837787603637191963573195, 7.14977691777917398809994079094, 7.16788345206229107744699076591, 7.53375307058247587357249366422, 8.459038134695047937455211668892, 8.867023978556716664989436402135, 9.620012553799449594182753328326, 9.842374305639781560858925177247, 10.36069300317884076015181233277, 11.04441630478008108341163433690, 11.05233437961054836011496367652, 11.79925645971910819215343496929, 12.16863657551990597683003049864

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