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Properties

Label 4-338688-1.1-c1e2-0-65
Degree $4$
Conductor $338688$
Sign $-1$
Analytic cond. $21.5950$
Root an. cond. $2.15570$
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  − 3-s + 2·5-s − 2·7-s + 9-s − 2·15-s + 6·17-s + 2·21-s − 6·25-s − 27-s − 4·35-s − 4·37-s − 10·41-s − 4·43-s + 2·45-s − 12·47-s − 3·49-s − 6·51-s − 8·59-s − 2·63-s + 4·67-s + 6·75-s + 12·79-s + 81-s − 4·83-s + 12·85-s − 2·89-s − 22·101-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.516·15-s + 1.45·17-s + 0.436·21-s − 6/5·25-s − 0.192·27-s − 0.676·35-s − 0.657·37-s − 1.56·41-s − 0.609·43-s + 0.298·45-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 1.04·59-s − 0.251·63-s + 0.488·67-s + 0.692·75-s + 1.35·79-s + 1/9·81-s − 0.439·83-s + 1.30·85-s − 0.211·89-s − 2.18·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338688 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(338688\)    =    \(2^{8} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(21.5950\)
Root analytic conductor: \(2.15570\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 338688,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
3$C_1$ \( 1 + T \)
7$C_2$ \( 1 + 2 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
11$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.11.a_c
13$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.13.a_ag
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) 2.17.ag_bq
19$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.19.a_abe
23$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.23.a_ba
29$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \) 2.29.a_be
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) 2.31.a_bu
37$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.37.e_o
41$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 10 T + p T^{2} ) \) 2.41.k_de
43$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.43.e_cw
47$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.47.m_dq
53$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \) 2.53.a_acg
59$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.59.i_cs
61$C_2^2$ \( 1 + 70 T^{2} + p^{2} T^{4} \) 2.61.a_cs
67$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.ae_bm
71$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \) 2.71.a_c
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \) 2.73.a_by
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.79.am_hi
83$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.83.e_cs
89$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) 2.89.c_cg
97$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.97.a_ek
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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

−8.394198173691313675274678569583, −8.101799801217331695896631501934, −7.60947383493433245341346620710, −6.90680275705825059714993956526, −6.63943032838070153707090868671, −6.06268117574169472565081211288, −5.78278146656308043215275589783, −5.14315867430249819520966908961, −4.91701865173131690987279015407, −3.93457430501747981709783061947, −3.45670944001019713706041716807, −2.94130919827521830364448199075, −1.94432058728612851558455956249, −1.41061215283782895602970449161, 0, 1.41061215283782895602970449161, 1.94432058728612851558455956249, 2.94130919827521830364448199075, 3.45670944001019713706041716807, 3.93457430501747981709783061947, 4.91701865173131690987279015407, 5.14315867430249819520966908961, 5.78278146656308043215275589783, 6.06268117574169472565081211288, 6.63943032838070153707090868671, 6.90680275705825059714993956526, 7.60947383493433245341346620710, 8.101799801217331695896631501934, 8.394198173691313675274678569583

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