Properties

Label 2-3332-1.1-c1-0-19
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  + 3·3-s − 4·5-s + 6·9-s + 11-s − 3·13-s − 12·15-s + 17-s + 2·19-s + 4·23-s + 11·25-s + 9·27-s − 8·31-s + 3·33-s + 8·37-s − 9·39-s + 10·43-s − 24·45-s + 10·47-s + 3·51-s + 3·53-s − 4·55-s + 6·57-s + 14·59-s + 8·61-s + 12·65-s − 10·67-s + 12·69-s + ⋯
L(s)  = 1  + 1.73·3-s − 1.78·5-s + 2·9-s + 0.301·11-s − 0.832·13-s − 3.09·15-s + 0.242·17-s + 0.458·19-s + 0.834·23-s + 11/5·25-s + 1.73·27-s − 1.43·31-s + 0.522·33-s + 1.31·37-s − 1.44·39-s + 1.52·43-s − 3.57·45-s + 1.45·47-s + 0.420·51-s + 0.412·53-s − 0.539·55-s + 0.794·57-s + 1.82·59-s + 1.02·61-s + 1.48·65-s − 1.22·67-s + 1.44·69-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.595555469\)
\(L(\frac12)\) \(\approx\) \(2.595555469\)
\(L(\frac{3}{2})\) 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 \( 1 - p T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 5 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−8.546135744533801374859937283605, −7.88145259888108741197353629005, −7.33219970595347634820696886920, −7.00911306004628352901119510439, −5.39416633360719924119943078408, −4.27940824929622244243786949106, −3.91279637895486486904498341505, −3.09224973436857989775104104718, −2.38679386712501414153520681691, −0.897265947653841331899552843846, 0.897265947653841331899552843846, 2.38679386712501414153520681691, 3.09224973436857989775104104718, 3.91279637895486486904498341505, 4.27940824929622244243786949106, 5.39416633360719924119943078408, 7.00911306004628352901119510439, 7.33219970595347634820696886920, 7.88145259888108741197353629005, 8.546135744533801374859937283605

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