Properties

Label 4-60372-1.1-c1e2-0-2
Degree $4$
Conductor $60372$
Sign $1$
Analytic cond. $3.84937$
Root an. cond. $1.40070$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 4-s + 2·7-s + 9-s + 12-s − 13-s + 16-s + 12·19-s + 2·21-s − 2·25-s + 27-s + 2·28-s − 13·31-s + 36-s − 37-s − 39-s − 4·43-s + 48-s − 11·49-s − 52-s + 12·57-s − 9·61-s + 2·63-s + 64-s + 3·67-s + 8·73-s − 2·75-s + ⋯
L(s)  = 1  + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s − 0.277·13-s + 1/4·16-s + 2.75·19-s + 0.436·21-s − 2/5·25-s + 0.192·27-s + 0.377·28-s − 2.33·31-s + 1/6·36-s − 0.164·37-s − 0.160·39-s − 0.609·43-s + 0.144·48-s − 1.57·49-s − 0.138·52-s + 1.58·57-s − 1.15·61-s + 0.251·63-s + 1/8·64-s + 0.366·67-s + 0.936·73-s − 0.230·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60372 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60372 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(60372\)    =    \(2^{2} \cdot 3^{3} \cdot 13 \cdot 43\)
Sign: $1$
Analytic conductor: \(3.84937\)
Root analytic conductor: \(1.40070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 60372,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.157061102\)
\(L(\frac12)\) \(\approx\) \(2.157061102\)
\(L(\frac{3}{2})\) 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$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
3$C_1$ \( 1 - T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
43$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 5 T + p T^{2} ) \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.5.a_c
7$C_2$ \( ( 1 - T + p T^{2} )^{2} \) 2.7.ac_p
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \) 2.11.a_ak
17$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.17.a_e
19$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \) 2.19.am_cw
23$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \) 2.23.a_q
29$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.29.a_abe
31$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.31.n_do
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) 2.37.b_abk
41$C_2^2$ \( 1 - 65 T^{2} + p^{2} T^{4} \) 2.41.a_acn
47$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.47.a_acx
53$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.53.a_cw
59$C_2^2$ \( 1 + 53 T^{2} + p^{2} T^{4} \) 2.59.a_cb
61$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.61.j_fa
67$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.67.ad_eu
71$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.71.a_acg
73$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.73.ai_gg
79$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.79.b_fq
83$C_2^2$ \( 1 - 79 T^{2} + p^{2} T^{4} \) 2.83.a_adb
89$C_2^2$ \( 1 + 100 T^{2} + p^{2} T^{4} \) 2.89.a_dw
97$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \) 2.97.ah_hm
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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

−9.777501113406351934009055227827, −9.493648241575057116357438112949, −9.039289090108830685873411859687, −8.327570677881300112866859978083, −7.77486166060591501181570454745, −7.48370955181385428827334518697, −7.08050499175654477128534742080, −6.32977909262771435244452247765, −5.48399758864567810994638812350, −5.23228730532936014427927373075, −4.50431917356803343709437795020, −3.43966460978132134694791482250, −3.26354491335369660228209682010, −2.12553193055181324599296544593, −1.40916995904576194686261217355, 1.40916995904576194686261217355, 2.12553193055181324599296544593, 3.26354491335369660228209682010, 3.43966460978132134694791482250, 4.50431917356803343709437795020, 5.23228730532936014427927373075, 5.48399758864567810994638812350, 6.32977909262771435244452247765, 7.08050499175654477128534742080, 7.48370955181385428827334518697, 7.77486166060591501181570454745, 8.327570677881300112866859978083, 9.039289090108830685873411859687, 9.493648241575057116357438112949, 9.777501113406351934009055227827

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