Properties

Label 4-3332e2-1.1-c0e2-0-9
Degree $4$
Conductor $11102224$
Sign $1$
Analytic cond. $2.76518$
Root an. cond. $1.28952$
Motivic weight $0$
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  + 2-s − 8-s + 9-s + 4·13-s − 16-s + 17-s + 18-s − 25-s + 4·26-s + 34-s − 50-s − 2·53-s + 64-s − 72-s − 2·89-s − 2·101-s − 4·104-s − 2·106-s + 4·117-s + 121-s + 127-s + 128-s + 131-s − 136-s + 137-s + 139-s − 144-s + ⋯
L(s)  = 1  + 2-s − 8-s + 9-s + 4·13-s − 16-s + 17-s + 18-s − 25-s + 4·26-s + 34-s − 50-s − 2·53-s + 64-s − 72-s − 2·89-s − 2·101-s − 4·104-s − 2·106-s + 4·117-s + 121-s + 127-s + 128-s + 131-s − 136-s + 137-s + 139-s − 144-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11102224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11102224\)    =    \(2^{4} \cdot 7^{4} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2.76518\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3332} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 11102224,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.885624111\)
\(L(\frac12)\) \(\approx\) \(2.885624111\)
\(L(1)\) 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$C_2$ \( 1 - T + T^{2} \)
7 \( 1 \)
17$C_2$ \( 1 - T + T^{2} \)
good3$C_2^2$ \( 1 - T^{2} + T^{4} \)
5$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_2^2$ \( 1 - T^{2} + T^{4} \)
13$C_1$ \( ( 1 - T )^{4} \)
19$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
23$C_2^2$ \( 1 - T^{2} + T^{4} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2^2$ \( 1 - T^{2} + T^{4} \)
37$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_2$ \( ( 1 + T^{2} )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2^2$ \( 1 - T^{2} + T^{4} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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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

−8.870431060253199522193022083744, −8.612420346822928727030383580695, −8.176187904380130979774471702948, −8.052473101106199130887091473846, −7.56244060526558704183467388904, −6.78421467486257145275805207514, −6.76629422866384744025966600858, −6.17516738020270335602904850429, −5.95236142671556961796155853963, −5.62264660835330541231651592472, −5.36332984954896989788059762674, −4.47887448849527279390066982580, −4.43172519368160728415277307891, −3.77334980129548481202183683420, −3.72845319858894840005191816088, −3.24897868620786090643291514224, −2.87969008200034055028423249611, −1.87312827483466508110981069627, −1.43444400753966688670149187750, −1.02368798150210056960391418729, 1.02368798150210056960391418729, 1.43444400753966688670149187750, 1.87312827483466508110981069627, 2.87969008200034055028423249611, 3.24897868620786090643291514224, 3.72845319858894840005191816088, 3.77334980129548481202183683420, 4.43172519368160728415277307891, 4.47887448849527279390066982580, 5.36332984954896989788059762674, 5.62264660835330541231651592472, 5.95236142671556961796155853963, 6.17516738020270335602904850429, 6.76629422866384744025966600858, 6.78421467486257145275805207514, 7.56244060526558704183467388904, 8.052473101106199130887091473846, 8.176187904380130979774471702948, 8.612420346822928727030383580695, 8.870431060253199522193022083744

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