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Properties

Label 2-3520-220.43-c0-0-3
Degree $2$
Conductor $3520$
Sign $0.999 + 0.0299i$
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  + (0.366 − 0.366i)3-s + (−0.866 + 0.5i)5-s + 0.732i·9-s − i·11-s + (−0.133 + 0.5i)15-s + (1.36 − 1.36i)23-s + (0.499 − 0.866i)25-s + (0.633 + 0.633i)27-s + i·31-s + (−0.366 − 0.366i)33-s + (0.366 − 0.366i)37-s + (−0.366 − 0.633i)45-s + (1 + i)47-s + i·49-s + (1 + i)53-s + ⋯
L(s)  = 1  + (0.366 − 0.366i)3-s + (−0.866 + 0.5i)5-s + 0.732i·9-s − i·11-s + (−0.133 + 0.5i)15-s + (1.36 − 1.36i)23-s + (0.499 − 0.866i)25-s + (0.633 + 0.633i)27-s + i·31-s + (−0.366 − 0.366i)33-s + (0.366 − 0.366i)37-s + (−0.366 − 0.633i)45-s + (1 + i)47-s + i·49-s + (1 + i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.999 + 0.0299i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.999 + 0.0299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.254892149\)
\(L(\frac12)\) \(\approx\) \(1.254892149\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + iT \)
good3 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - iT - T^{2} \)
37 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1 - i)T + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.36 - 1.36i)T + iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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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

−8.563024032167143857703740813717, −8.070911599258350399288473287016, −7.27211143591740624472733861088, −6.76540062329514853922926615823, −5.80833310190659282912037491624, −4.86457783809775548302963082359, −4.06437807779783967174474678774, −3.01741973512021167931426175034, −2.54894535252613818460743548308, −1.00246733385171641448414528238, 0.972635815056986707693128646543, 2.34942974223841877642217657175, 3.52911326189471457829587341091, 3.95790545309948806474679381960, 4.87543184082141357472029269702, 5.54662003530230024968124048549, 6.82139458468136578600172912307, 7.22727067934218682571770720760, 8.130727219521576512932279233143, 8.751518245299808202750864330835

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