Properties

Label 2-3520-1.1-c1-0-11
Degree $2$
Conductor $3520$
Sign $1$
Analytic cond. $28.1073$
Root an. cond. $5.30163$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·3-s + 5-s − 2·7-s + 5.00·9-s − 11-s + 6.82·13-s − 2.82·15-s + 1.17·17-s + 5.65·21-s + 2.82·23-s + 25-s − 5.65·27-s − 7.65·29-s + 2.82·33-s − 2·35-s − 3.65·37-s − 19.3·39-s + 6·41-s + 6·43-s + 5.00·45-s − 2.82·47-s − 3·49-s − 3.31·51-s − 0.343·53-s − 55-s + 9.65·59-s − 13.3·61-s + ⋯
L(s)  = 1  − 1.63·3-s + 0.447·5-s − 0.755·7-s + 1.66·9-s − 0.301·11-s + 1.89·13-s − 0.730·15-s + 0.284·17-s + 1.23·21-s + 0.589·23-s + 0.200·25-s − 1.08·27-s − 1.42·29-s + 0.492·33-s − 0.338·35-s − 0.601·37-s − 3.09·39-s + 0.937·41-s + 0.914·43-s + 0.745·45-s − 0.412·47-s − 0.428·49-s − 0.464·51-s − 0.0471·53-s − 0.134·55-s + 1.25·59-s − 1.70·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $1$
Analytic conductor: \(28.1073\)
Root analytic conductor: \(5.30163\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9979520420\)
\(L(\frac12)\) \(\approx\) \(0.9979520420\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 - 6.82T + 13T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 0.343T + 53T^{2} \)
59 \( 1 - 9.65T + 59T^{2} \)
61 \( 1 + 13.3T + 61T^{2} \)
67 \( 1 - 4.48T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 6.82T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 9.31T + 89T^{2} \)
97 \( 1 + 7.65T + 97T^{2} \)
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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

−8.724492246229974815332020431449, −7.60462515660041835347924566117, −6.82904431082744527442095660219, −6.02745271972089421507969514298, −5.87484216211964272315935782236, −5.00586822143616922746744999482, −4.01674866907012371769941478268, −3.16511995193899276948389951815, −1.66409038449636696386426825227, −0.66549029873207549241152803058, 0.66549029873207549241152803058, 1.66409038449636696386426825227, 3.16511995193899276948389951815, 4.01674866907012371769941478268, 5.00586822143616922746744999482, 5.87484216211964272315935782236, 6.02745271972089421507969514298, 6.82904431082744527442095660219, 7.60462515660041835347924566117, 8.724492246229974815332020431449

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