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Properties

Label 4-107653-1.1-c1e2-0-1
Degree $4$
Conductor $107653$
Sign $-1$
Analytic cond. $6.86404$
Root an. cond. $1.61862$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·9-s − 13-s − 4·16-s + 8·17-s + 6·23-s − 25-s − 10·29-s − 2·43-s + 49-s − 18·53-s − 20·61-s + 6·79-s + 27·81-s − 28·101-s − 8·103-s − 8·107-s − 6·113-s + 6·117-s + 14·121-s + 127-s + 131-s + 137-s + 139-s + 24·144-s + 149-s + 151-s − 48·153-s + ⋯
L(s)  = 1  − 2·9-s − 0.277·13-s − 16-s + 1.94·17-s + 1.25·23-s − 1/5·25-s − 1.85·29-s − 0.304·43-s + 1/7·49-s − 2.47·53-s − 2.56·61-s + 0.675·79-s + 3·81-s − 2.78·101-s − 0.788·103-s − 0.773·107-s − 0.564·113-s + 0.554·117-s + 1.27·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2·144-s + 0.0819·149-s + 0.0813·151-s − 3.88·153-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 107653 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 107653 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(107653\)    =    \(7^{2} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(6.86404\)
Root analytic conductor: \(1.61862\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 107653,\ (\ :1/2, 1/2),\ -1)\)

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 $\mathbf{F}_p$
bad7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) 2.2.a_a
3$C_2$ \( ( 1 + p T^{2} )^{2} \) 2.3.a_g
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.5.a_b
11$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.11.a_ao
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \) 2.17.ai_by
19$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) 2.19.a_n
23$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.23.ag_cd
29$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \) 2.29.k_df
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.31.a_cb
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.37.a_cg
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.41.a_bu
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \) 2.43.c_dj
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.47.a_bt
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \) 2.53.s_hf
59$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.59.a_cc
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \) 2.61.u_io
67$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.67.a_du
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.71.a_da
73$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) 2.73.a_ax
79$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \) 2.79.ag_gl
83$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) 2.83.a_ach
89$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) 2.89.a_gn
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.97.a_fp
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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

−9.207834379220788354016903344416, −8.998346772410808154035420365472, −8.157183950347712692726002057248, −7.88305646625584781075065151406, −7.43563929744309328324545511818, −6.69002626594602126625681974705, −6.16699839705098439022921833193, −5.57652565474586077264622741418, −5.27883313770925671699270459709, −4.66814317251753947271974703426, −3.69113279097650649608530641419, −3.11744276397134147631028514628, −2.68417503048537603951806988505, −1.57613473105511884594471340976, 0, 1.57613473105511884594471340976, 2.68417503048537603951806988505, 3.11744276397134147631028514628, 3.69113279097650649608530641419, 4.66814317251753947271974703426, 5.27883313770925671699270459709, 5.57652565474586077264622741418, 6.16699839705098439022921833193, 6.69002626594602126625681974705, 7.43563929744309328324545511818, 7.88305646625584781075065151406, 8.157183950347712692726002057248, 8.998346772410808154035420365472, 9.207834379220788354016903344416

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