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Properties

Label 2-3520-220.43-c0-0-4
Degree $2$
Conductor $3520$
Sign $0.473 + 0.880i$
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  + (1.36 − 1.36i)3-s + (0.866 + 0.5i)5-s − 2.73i·9-s + i·11-s + (1.86 − 0.499i)15-s + (0.366 − 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s − i·31-s + (1.36 + 1.36i)33-s + (−1.36 + 1.36i)37-s + (1.36 − 2.36i)45-s + (−1 − i)47-s + i·49-s + (1 + i)53-s + ⋯
L(s)  = 1  + (1.36 − 1.36i)3-s + (0.866 + 0.5i)5-s − 2.73i·9-s + i·11-s + (1.86 − 0.499i)15-s + (0.366 − 0.366i)23-s + (0.499 + 0.866i)25-s + (−2.36 − 2.36i)27-s − i·31-s + (1.36 + 1.36i)33-s + (−1.36 + 1.36i)37-s + (1.36 − 2.36i)45-s + (−1 − i)47-s + i·49-s + (1 + i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.473 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.473 + 0.880i$
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: \((2,\ 3520,\ (\ :0),\ 0.473 + 0.880i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.350444957\)
\(L(\frac12)\) \(\approx\) \(2.350444957\)
\(L(1)\) 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 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
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 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + iT - T^{2} \)
97 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
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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

−8.603841592470261933991787431209, −7.77192170679599747965670427612, −7.18942038888676834124367048552, −6.62021304312556647701840143222, −5.99453055073374346423893448112, −4.78769822622073517367405033268, −3.59005700980531934433078715140, −2.79548847338052011411207802692, −2.09866558210741288712812959879, −1.38517999427716315462962919478, 1.65902306458581025874890998581, 2.64408239830224144655036553862, 3.39698960806155605147626714270, 4.11500314691135427787365194067, 5.16263418663908602955247717331, 5.43359695500295790928271136208, 6.66333111791542193348726367509, 7.72493328897334356632167082815, 8.585754475757338544277759303444, 8.787031487878822493297407499253

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