Properties

Label 2-3332-1.1-c1-0-15
Degree $2$
Conductor $3332$
Sign $1$
Analytic cond. $26.6061$
Root an. cond. $5.15811$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·9-s − 5·11-s + 5·13-s + 17-s + 6·19-s + 4·23-s − 5·25-s − 5·27-s + 4·29-s − 5·33-s + 8·37-s + 5·39-s − 4·41-s − 6·43-s + 6·47-s + 51-s + 11·53-s + 6·57-s − 10·59-s + 10·67-s + 4·69-s + 9·71-s − 4·73-s − 5·75-s + 9·79-s + 81-s + ⋯
L(s)  = 1  + 0.577·3-s − 2/3·9-s − 1.50·11-s + 1.38·13-s + 0.242·17-s + 1.37·19-s + 0.834·23-s − 25-s − 0.962·27-s + 0.742·29-s − 0.870·33-s + 1.31·37-s + 0.800·39-s − 0.624·41-s − 0.914·43-s + 0.875·47-s + 0.140·51-s + 1.51·53-s + 0.794·57-s − 1.30·59-s + 1.22·67-s + 0.481·69-s + 1.06·71-s − 0.468·73-s − 0.577·75-s + 1.01·79-s + 1/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $1$
Analytic conductor: \(26.6061\)
Root analytic conductor: \(5.15811\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3332} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.114165793\)
\(L(\frac12)\) \(\approx\) \(2.114165793\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
17 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 9 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.525660609948937391421088526160, −7.955765237844670697954364389904, −7.41172189993609815460068814708, −6.26466643131549230083126258040, −5.58418891633662142237727958169, −4.92349843099561163420302937605, −3.65727267712361259907847714013, −3.07998893972865726925809645422, −2.22933027974968160366994785789, −0.843059013728295182413899131278, 0.843059013728295182413899131278, 2.22933027974968160366994785789, 3.07998893972865726925809645422, 3.65727267712361259907847714013, 4.92349843099561163420302937605, 5.58418891633662142237727958169, 6.26466643131549230083126258040, 7.41172189993609815460068814708, 7.955765237844670697954364389904, 8.525660609948937391421088526160

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