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Properties

Label 2-3520-220.43-c0-0-5
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

\(L(\frac{1}{2})\) \(\approx\) \(1.702118650\)
\(L(\frac12)\) \(\approx\) \(1.702118650\)
\(L(1)\) 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

−8.373121607941518872698173430142, −7.71306184159860454152181016566, −7.51366561642598946347535082897, −5.98997272316989416539037392683, −5.89491416418119443838896801987, −4.52294321957145122556567156863, −3.75576818628344640257844640158, −2.74235442698613735742205764445, −1.87878290293833525195146886545, −0.884116654868285036603463869265, 1.95986582775280986915833989711, 2.75972577348477865730905137322, 3.49148663243138970128139299962, 4.26871364490257042912855529142, 4.93672759219392977097245953046, 6.12002223052367527014788395765, 6.87904312531357524354101443112, 7.57432548371909978402923575997, 8.426804945928555706539348257693, 8.959413830882432028919420140206

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