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Properties

Label 4-1707552-1.1-c1e2-0-49
Degree $4$
Conductor $1707552$
Sign $-1$
Analytic cond. $108.874$
Root an. cond. $3.23021$
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-s + 4-s + 8-s − 9-s + 2·11-s + 16-s − 18-s + 2·22-s + 12·23-s + 2·25-s − 16·29-s + 32-s − 36-s + 4·37-s − 12·43-s + 2·44-s + 12·46-s − 7·49-s + 2·50-s − 16·53-s − 16·58-s + 64-s + 8·67-s − 20·71-s − 72-s + 4·74-s − 8·79-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1/3·9-s + 0.603·11-s + 1/4·16-s − 0.235·18-s + 0.426·22-s + 2.50·23-s + 2/5·25-s − 2.97·29-s + 0.176·32-s − 1/6·36-s + 0.657·37-s − 1.82·43-s + 0.301·44-s + 1.76·46-s − 49-s + 0.282·50-s − 2.19·53-s − 2.10·58-s + 1/8·64-s + 0.977·67-s − 2.37·71-s − 0.117·72-s + 0.464·74-s − 0.900·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1707552 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1707552\)    =    \(2^{5} \cdot 3^{2} \cdot 7^{2} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(108.874\)
Root analytic conductor: \(3.23021\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 1707552,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad2$C_1$ \( 1 - T \)
3$C_2$ \( 1 + T^{2} \)
7$C_2$ \( 1 + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \) 2.5.a_ac
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.17.a_s
19$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.19.a_aba
23$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.23.am_da
29$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.29.q_eo
31$C_2^2$ \( 1 - 14 T^{2} + p^{2} T^{4} \) 2.31.a_ao
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.37.ae_o
41$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.41.a_abu
43$C_2$$\times$$C_2$ \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.43.m_eo
47$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \) 2.47.a_abu
53$C_2$$\times$$C_2$ \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.53.q_gk
59$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \) 2.59.a_aw
61$C_2^2$ \( 1 + 42 T^{2} + p^{2} T^{4} \) 2.61.a_bq
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.67.ai_di
71$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.71.u_je
73$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \) 2.73.a_k
79$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.79.i_eg
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.a_w
89$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.89.a_eg
97$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \) 2.97.a_eg
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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

−7.57200618119974847377060754081, −7.01574882776360207559256114650, −6.78199091838135607814972576394, −6.40703127490508891016399542314, −5.74931006280756116853522624304, −5.45560031739897703847254496372, −4.98596748270000360171806634285, −4.61085454118806496720369005029, −3.99635328968641356940989759802, −3.49602724176109222205625883774, −3.06678583033327235392288788502, −2.65090828519084735767923985386, −1.65504222427944030339450178016, −1.37940438394703721991360517634, 0, 1.37940438394703721991360517634, 1.65504222427944030339450178016, 2.65090828519084735767923985386, 3.06678583033327235392288788502, 3.49602724176109222205625883774, 3.99635328968641356940989759802, 4.61085454118806496720369005029, 4.98596748270000360171806634285, 5.45560031739897703847254496372, 5.74931006280756116853522624304, 6.40703127490508891016399542314, 6.78199091838135607814972576394, 7.01574882776360207559256114650, 7.57200618119974847377060754081

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