Properties

Label 2-296208-1.1-c1-0-126
Degree $2$
Conductor $296208$
Sign $1$
Analytic cond. $2365.23$
Root an. cond. $48.6336$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 2·7-s − 17-s − 6·23-s + 11·25-s − 2·29-s − 4·31-s + 8·35-s + 2·37-s − 6·41-s − 4·43-s − 6·47-s − 3·49-s − 8·53-s + 8·59-s + 8·61-s + 4·67-s − 6·71-s − 10·73-s − 6·79-s + 4·83-s + 4·85-s + 14·89-s + 14·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  − 1.78·5-s − 0.755·7-s − 0.242·17-s − 1.25·23-s + 11/5·25-s − 0.371·29-s − 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.875·47-s − 3/7·49-s − 1.09·53-s + 1.04·59-s + 1.02·61-s + 0.488·67-s − 0.712·71-s − 1.17·73-s − 0.675·79-s + 0.439·83-s + 0.433·85-s + 1.48·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(296208\)    =    \(2^{4} \cdot 3^{2} \cdot 11^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(2365.23\)
Root analytic conductor: \(48.6336\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{296208} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 296208,\ (\ :1/2),\ 1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
17 \( 1 + T \)
good5 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
13 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 8 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.99167328392685, −12.81947787095293, −12.08066048109277, −11.67901157634794, −11.58914348992697, −10.94547711902342, −10.44082439831525, −9.961334014469031, −9.537137326512442, −8.854971216916289, −8.489803680833676, −8.071928525970368, −7.515429132269153, −7.263787828192149, −6.591226659768300, −6.276939909321646, −5.620787068139221, −4.851818094947785, −4.608502307834513, −3.861983297065919, −3.505648426840604, −3.265129753764886, −2.410933418696835, −1.786923543377196, −0.9159107958654783, 0, 0, 0.9159107958654783, 1.786923543377196, 2.410933418696835, 3.265129753764886, 3.505648426840604, 3.861983297065919, 4.608502307834513, 4.851818094947785, 5.620787068139221, 6.276939909321646, 6.591226659768300, 7.263787828192149, 7.515429132269153, 8.071928525970368, 8.489803680833676, 8.854971216916289, 9.537137326512442, 9.961334014469031, 10.44082439831525, 10.94547711902342, 11.58914348992697, 11.67901157634794, 12.08066048109277, 12.81947787095293, 12.99167328392685

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