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Properties

Label 2-3520-440.149-c0-0-3
Degree $2$
Conductor $3520$
Sign $-0.434 + 0.900i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.309 − 0.951i)5-s + (0.951 − 0.690i)7-s + (−0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (−1.80 + 0.587i)13-s + (1.53 + 1.11i)19-s − 1.61i·23-s + (−0.809 − 0.587i)25-s + (−0.363 − 1.11i)35-s + (−0.5 + 0.363i)37-s + (−0.690 + 0.951i)41-s − 0.999·45-s + (0.363 − 0.5i)47-s + (0.118 − 0.363i)49-s + (−0.5 − 1.53i)53-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)5-s + (0.951 − 0.690i)7-s + (−0.309 − 0.951i)9-s + (−0.587 − 0.809i)11-s + (−1.80 + 0.587i)13-s + (1.53 + 1.11i)19-s − 1.61i·23-s + (−0.809 − 0.587i)25-s + (−0.363 − 1.11i)35-s + (−0.5 + 0.363i)37-s + (−0.690 + 0.951i)41-s − 0.999·45-s + (0.363 − 0.5i)47-s + (0.118 − 0.363i)49-s + (−0.5 − 1.53i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.434 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.434 + 0.900i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.434 + 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.168434031\)
\(L(\frac12)\) \(\approx\) \(1.168434031\)
\(L(1)\) 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 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.587 + 0.809i)T \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + 1.61iT - T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - 1.61T + T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−8.398184041801162740642601044531, −7.941949097565649792405650482833, −7.11723273151184885025095218058, −6.23241740294435405812712015713, −5.25361022856717361252459816774, −4.87517483068462296663195150130, −3.97004435382366611396061114442, −2.91882231587356650098016040788, −1.76636677620376176427605368020, −0.65514776815750838012169449112, 1.89741411299580832486047724925, 2.48780750996050953198178367101, 3.20559759854973202297456463790, 4.78658345028387825246945887601, 5.20131565453995587297396434528, 5.69473013127275247080471398722, 7.21117070031136654300419357882, 7.39614930710868787415647688226, 7.959781160173283057690908338185, 9.123490768744066148528478445193

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