Properties

Label 4-390e2-1.1-c1e2-0-12
Degree $4$
Conductor $152100$
Sign $1$
Analytic cond. $9.69802$
Root an. cond. $1.76469$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4-s + 4·5-s − 9-s + 4·11-s + 16-s + 8·19-s − 4·20-s + 11·25-s − 8·29-s + 16·31-s + 36-s − 12·41-s − 4·44-s − 4·45-s + 10·49-s + 16·55-s − 20·59-s − 28·61-s − 64-s − 8·71-s − 8·76-s + 16·79-s + 4·80-s + 81-s − 12·89-s + 32·95-s − 4·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s − 1/3·9-s + 1.20·11-s + 1/4·16-s + 1.83·19-s − 0.894·20-s + 11/5·25-s − 1.48·29-s + 2.87·31-s + 1/6·36-s − 1.87·41-s − 0.603·44-s − 0.596·45-s + 10/7·49-s + 2.15·55-s − 2.60·59-s − 3.58·61-s − 1/8·64-s − 0.949·71-s − 0.917·76-s + 1.80·79-s + 0.447·80-s + 1/9·81-s − 1.27·89-s + 3.28·95-s − 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(152100\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(9.69802\)
Root analytic conductor: \(1.76469\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{390} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 152100,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.279236862\)
\(L(\frac12)\) \(\approx\) \(2.279236862\)
\(L(\frac{3}{2})\) 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)$
bad2$C_2$ \( 1 + T^{2} \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
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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

−11.67929741531403913953414891160, −11.03988785741767423611475136287, −10.40349342832384002536881095366, −10.15182428810637106566355532472, −9.559245089767598976866079494335, −9.428101240143841676202585748156, −8.794398656612434567960187662607, −8.720742915198779871262138883141, −7.61178114907466501403572635929, −7.52845205699816026568328387423, −6.57053871516389154097696472168, −6.14513308344513577840423000173, −6.00873995796246908305735998507, −5.07192883066494602073126526013, −4.96935059804846759672691954704, −4.13125857353407705589556702147, −3.21911290967116870543334881841, −2.85808539939422729727876459943, −1.74012033866586859306908382209, −1.18946337852103836515029025778, 1.18946337852103836515029025778, 1.74012033866586859306908382209, 2.85808539939422729727876459943, 3.21911290967116870543334881841, 4.13125857353407705589556702147, 4.96935059804846759672691954704, 5.07192883066494602073126526013, 6.00873995796246908305735998507, 6.14513308344513577840423000173, 6.57053871516389154097696472168, 7.52845205699816026568328387423, 7.61178114907466501403572635929, 8.720742915198779871262138883141, 8.794398656612434567960187662607, 9.428101240143841676202585748156, 9.559245089767598976866079494335, 10.15182428810637106566355532472, 10.40349342832384002536881095366, 11.03988785741767423611475136287, 11.67929741531403913953414891160

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