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Properties

Label 2-3536-884.87-c0-0-1
Degree $2$
Conductor $3536$
Sign $0.207 + 0.978i$
Analytic cond. $1.76469$
Root an. cond. $1.32841$
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  + (0.0999 + 0.241i)5-s + (−0.965 − 0.258i)9-s + (−0.258 − 0.965i)13-s + (−0.866 − 0.5i)17-s + (0.658 − 0.658i)25-s + (1.20 + 0.158i)29-s + (0.258 − 1.96i)37-s + (−0.741 − 0.965i)41-s + (−0.0340 − 0.258i)45-s + (0.965 − 0.258i)49-s + (−1.36 − 1.36i)53-s + (−0.258 + 0.0340i)61-s + (0.207 − 0.158i)65-s + (0.465 + 1.12i)73-s + (0.866 + 0.499i)81-s + ⋯
L(s)  = 1  + (0.0999 + 0.241i)5-s + (−0.965 − 0.258i)9-s + (−0.258 − 0.965i)13-s + (−0.866 − 0.5i)17-s + (0.658 − 0.658i)25-s + (1.20 + 0.158i)29-s + (0.258 − 1.96i)37-s + (−0.741 − 0.965i)41-s + (−0.0340 − 0.258i)45-s + (0.965 − 0.258i)49-s + (−1.36 − 1.36i)53-s + (−0.258 + 0.0340i)61-s + (0.207 − 0.158i)65-s + (0.465 + 1.12i)73-s + (0.866 + 0.499i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $0.207 + 0.978i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (1855, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.207 + 0.978i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9443651793\)
\(L(\frac12)\) \(\approx\) \(0.9443651793\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.258 + 0.965i)T \)
17 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (-0.0999 - 0.241i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.965 + 0.258i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T^{2} \)
29 \( 1 + (-1.20 - 0.158i)T + (0.965 + 0.258i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.258 + 1.96i)T + (-0.965 - 0.258i)T^{2} \)
41 \( 1 + (0.741 + 0.965i)T + (-0.258 + 0.965i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.258 - 0.0340i)T + (0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.258 - 0.965i)T^{2} \)
73 \( 1 + (-0.465 - 1.12i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 + (-0.707 - 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.12 - 1.46i)T + (-0.258 - 0.965i)T^{2} \)
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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

−8.606210343478225805910147644943, −7.943847983823700132206262411505, −7.01858406252503976906820731078, −6.40392981765973156060667917073, −5.54323621411907037838794657685, −4.92245695511170719568927058072, −3.84268579854527957074948798397, −2.91050971084434676905634644481, −2.27499259986445452651781265189, −0.55107687937305756422312457387, 1.41747096675218103277368120575, 2.50739367818359576664717188926, 3.31484250648445856537990156752, 4.57726746249380324626119107564, 4.89635679851564625446219904984, 6.11615763794309558753352887659, 6.49746354876623205398045858265, 7.47127031004456854263842896218, 8.345617735429863487700285892259, 8.808094265607341412803577033439

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