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Properties

Label 2-3520-55.54-c0-0-1
Degree $2$
Conductor $3520$
Sign $-0.5 + 0.866i$
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  + 1.73i·3-s + (−0.5 + 0.866i)5-s − 1.99·9-s − 11-s + (−1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 − 0.866i)25-s − 1.73i·27-s − 31-s − 1.73i·33-s − 1.73i·37-s + (0.999 − 1.73i)45-s − 49-s + (0.5 − 0.866i)55-s − 59-s + ⋯
L(s)  = 1  + 1.73i·3-s + (−0.5 + 0.866i)5-s − 1.99·9-s − 11-s + (−1.49 − 0.866i)15-s + 1.73i·23-s + (−0.499 − 0.866i)25-s − 1.73i·27-s − 31-s − 1.73i·33-s − 1.73i·37-s + (0.999 − 1.73i)45-s − 49-s + (0.5 − 0.866i)55-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.5 + 0.866i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5097604018\)
\(L(\frac12)\) \(\approx\) \(0.5097604018\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + T \)
good3 \( 1 - 1.73iT - T^{2} \)
7 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - 1.73iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + 1.73iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.428504491463141679516085083156, −8.650368751680316502717473649803, −7.75819046476505591318378039814, −7.21934500546732208509083743260, −5.95882370993506645310466065945, −5.40530249119367340094434705633, −4.60176857958982066703390468480, −3.69501106849078729635575909196, −3.30190917640204802683499719936, −2.28477131878187930013813737081, 0.28948903540452816138240000563, 1.44321629673653656317330822670, 2.36846965196186028614873395685, 3.29353579526678005725077686489, 4.62107298996191751137655508229, 5.27416466078034725385693887564, 6.22527588521186774159283290465, 6.80635429246482385375611906903, 7.72157126202704508272933977698, 8.066362191758170309427956179372

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