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Properties

Label 4-2376e2-1.1-c1e2-0-16
Degree $4$
Conductor $5645376$
Sign $-1$
Analytic cond. $359.954$
Root an. cond. $4.35573$
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  + 2·5-s + 4·11-s − 12·23-s − 6·25-s + 2·31-s − 10·37-s + 4·47-s + 49-s + 20·53-s + 8·55-s − 4·59-s − 6·67-s + 12·71-s − 16·89-s + 14·97-s + 22·103-s − 22·113-s − 24·115-s + 5·121-s − 22·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 4·155-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.20·11-s − 2.50·23-s − 6/5·25-s + 0.359·31-s − 1.64·37-s + 0.583·47-s + 1/7·49-s + 2.74·53-s + 1.07·55-s − 0.520·59-s − 0.733·67-s + 1.42·71-s − 1.69·89-s + 1.42·97-s + 2.16·103-s − 2.06·113-s − 2.23·115-s + 5/11·121-s − 1.96·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.321·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5645376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5645376\)    =    \(2^{6} \cdot 3^{6} \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(359.954\)
Root analytic conductor: \(4.35573\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 5645376,\ (\ :1/2, 1/2),\ -1)\)

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 \( 1 \)
11$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) 2.5.ac_k
7$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \) 2.7.a_ab
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \) 2.13.a_s
17$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \) 2.17.a_abf
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \) 2.19.a_g
23$C_2$$\times$$C_2$ \( ( 1 + 3 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.23.m_cv
29$C_2^2$ \( 1 + 41 T^{2} + p^{2} T^{4} \) 2.29.a_bp
31$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.31.ac_cc
37$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 9 T + p T^{2} ) \) 2.37.k_df
41$C_2^2$ \( 1 + 37 T^{2} + p^{2} T^{4} \) 2.41.a_bl
43$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \) 2.43.a_an
47$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - T + p T^{2} ) \) 2.47.ae_dt
53$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \) 2.53.au_hy
59$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) 2.59.e_dt
61$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \) 2.61.a_ba
67$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.67.g_eo
71$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) 2.71.am_gs
73$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \) 2.73.a_cw
79$C_2^2$ \( 1 + 75 T^{2} + p^{2} T^{4} \) 2.79.a_cx
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \) 2.83.a_ady
89$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) 2.89.q_fm
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \) 2.97.ao_jj
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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

−6.98677245051625828825830843322, −6.54223999291913242941327330836, −6.27380165312445507248936845953, −5.82108474149237791263855008659, −5.56609872329202499047326229627, −5.19812630664501953142218654307, −4.38044858593097489967291482516, −4.14471956881028489130418910320, −3.74202563595538339907128216360, −3.34150325635290105487032453449, −2.46885933219614516342241696596, −2.06472568382136776326294570126, −1.76125012439458407724842469813, −1.01497262312281262820879941968, 0, 1.01497262312281262820879941968, 1.76125012439458407724842469813, 2.06472568382136776326294570126, 2.46885933219614516342241696596, 3.34150325635290105487032453449, 3.74202563595538339907128216360, 4.14471956881028489130418910320, 4.38044858593097489967291482516, 5.19812630664501953142218654307, 5.56609872329202499047326229627, 5.82108474149237791263855008659, 6.27380165312445507248936845953, 6.54223999291913242941327330836, 6.98677245051625828825830843322

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