Properties

Label 4-70e2-1.1-c3e2-0-1
Degree $4$
Conductor $4900$
Sign $1$
Analytic cond. $17.0580$
Root an. cond. $2.03227$
Motivic weight $3$
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  − 4·4-s + 20·5-s + 5·9-s − 74·11-s + 16·16-s + 216·19-s − 80·20-s + 275·25-s + 498·29-s − 268·31-s − 20·36-s + 412·41-s + 296·44-s + 100·45-s − 49·49-s − 1.48e3·55-s + 4·59-s − 1.88e3·61-s − 64·64-s + 912·71-s − 864·76-s + 2.47e3·79-s + 320·80-s − 704·81-s + 440·89-s + 4.32e3·95-s − 370·99-s + ⋯
L(s)  = 1  − 1/2·4-s + 1.78·5-s + 5/27·9-s − 2.02·11-s + 1/4·16-s + 2.60·19-s − 0.894·20-s + 11/5·25-s + 3.18·29-s − 1.55·31-s − 0.0925·36-s + 1.56·41-s + 1.01·44-s + 0.331·45-s − 1/7·49-s − 3.62·55-s + 0.00882·59-s − 3.94·61-s − 1/8·64-s + 1.52·71-s − 1.30·76-s + 3.52·79-s + 0.447·80-s − 0.965·81-s + 0.524·89-s + 4.66·95-s − 0.375·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(17.0580\)
Root analytic conductor: \(2.03227\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{70} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 4900,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.172243306\)
\(L(\frac12)\) \(\approx\) \(2.172243306\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5$C_2$ \( 1 - 4 p T + p^{3} T^{2} \)
7$C_2$ \( 1 + p^{2} T^{2} \)
good3$C_2^2$ \( 1 - 5 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 37 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 1793 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 8145 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 108 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 19434 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 249 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 134 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 10250 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 206 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 17638 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 125277 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 297718 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 2 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 940 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 590290 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 456 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 355534 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 - 1239 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 960390 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 220 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 712321 T^{2} + p^{6} T^{4} \)
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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

−14.95874643872354539388060303324, −13.90514877564448857311366002736, −13.69891548369657476228828402675, −12.75209081584646640439851691702, −12.66087495841957018935546270111, −11.83451835575880977850606259165, −10.82667295576835483070048417340, −10.39445098877789195841852273401, −10.06365754421865754019791259563, −9.209710868199050433289692142871, −9.158416182249837633016151765760, −7.83490811575240044174261710539, −7.68636113125550124844878375250, −6.53912637881253195178942533153, −5.85196397332882706266720027416, −5.05328577804960084359990535266, −4.96533861786912371008243909024, −3.16164617336304600474876871373, −2.49723879914460301259313473329, −1.08181718597291226594940416913, 1.08181718597291226594940416913, 2.49723879914460301259313473329, 3.16164617336304600474876871373, 4.96533861786912371008243909024, 5.05328577804960084359990535266, 5.85196397332882706266720027416, 6.53912637881253195178942533153, 7.68636113125550124844878375250, 7.83490811575240044174261710539, 9.158416182249837633016151765760, 9.209710868199050433289692142871, 10.06365754421865754019791259563, 10.39445098877789195841852273401, 10.82667295576835483070048417340, 11.83451835575880977850606259165, 12.66087495841957018935546270111, 12.75209081584646640439851691702, 13.69891548369657476228828402675, 13.90514877564448857311366002736, 14.95874643872354539388060303324

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