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Properties

Label 2-3520-220.43-c0-0-2
Degree $2$
Conductor $3520$
Sign $0.525 - 0.850i$
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  + (−1 + i)3-s − i·5-s − i·9-s + i·11-s + (1 + i)15-s + (1 − i)23-s − 25-s + 2i·31-s + (−1 − i)33-s + (1 − i)37-s − 45-s + (−1 − i)47-s + i·49-s + (1 + i)53-s + 55-s + ⋯
L(s)  = 1  + (−1 + i)3-s − i·5-s − i·9-s + i·11-s + (1 + i)15-s + (1 − i)23-s − 25-s + 2i·31-s + (−1 − i)33-s + (1 − i)37-s − 45-s + (−1 − i)47-s + i·49-s + (1 + i)53-s + 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.525 - 0.850i$
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: \((2,\ 3520,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8378255882\)
\(L(\frac12)\) \(\approx\) \(0.8378255882\)
\(L(1)\) 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 \( 1 + iT \)
11 \( 1 - iT \)
good3 \( 1 + (1 - i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (1 + i)T + iT^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 - 2T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 - i)T + iT^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−8.983651263066578699248280535524, −8.340665336207782983593030117635, −7.28253876286306897680148022606, −6.54756521990959785530555479478, −5.54886385713072231141570545662, −5.03923374169437802326606388547, −4.48868296024441416417342353279, −3.77902691935010940142362345294, −2.37593680088120376600397780146, −1.00505512605218439871049998977, 0.71991185524936184361735781908, 1.97071636239104149430192202420, 3.00679434651931286955593195946, 3.85940411730862075620330946856, 5.16018383966096923085247208034, 5.86815054128346574055124545238, 6.38313577704326211181587672714, 7.03587750557261876324079651547, 7.68763684065457736898393380940, 8.373441748415860299525204179931

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