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Properties

Label 2-3536-884.679-c0-0-0
Degree $2$
Conductor $3536$
Sign $-0.967 - 0.252i$
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  + (0.965 + 1.67i)3-s + (−0.965 + 1.67i)7-s + (−1.36 + 2.36i)9-s + (0.707 + 1.22i)11-s − i·13-s + (0.5 − 0.866i)17-s − 3.73·21-s + (−0.258 − 0.448i)23-s + 25-s − 3.34·27-s + 0.517·31-s + (−1.36 + 2.36i)33-s + (1.67 − 0.965i)39-s + (−1.36 − 2.36i)49-s + 1.93·51-s + ⋯
L(s)  = 1  + (0.965 + 1.67i)3-s + (−0.965 + 1.67i)7-s + (−1.36 + 2.36i)9-s + (0.707 + 1.22i)11-s − i·13-s + (0.5 − 0.866i)17-s − 3.73·21-s + (−0.258 − 0.448i)23-s + 25-s − 3.34·27-s + 0.517·31-s + (−1.36 + 2.36i)33-s + (1.67 − 0.965i)39-s + (−1.36 − 2.36i)49-s + 1.93·51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 - 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $-0.967 - 0.252i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (2447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ -0.967 - 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.558302509\)
\(L(\frac12)\) \(\approx\) \(1.558302509\)
\(L(1)\) 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 \( 1 + iT \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - 0.517T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.93T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.118002085420241806231333140623, −8.686130871414107780284755473165, −7.929594076382926730634180576092, −6.87370731414286421511266877633, −5.81839819539141793906657651501, −5.16598289901299921553467033559, −4.51262331948524020690471550531, −3.48280674943275592491296368924, −2.86072844130768239840567170760, −2.26677234066415533876620655986, 0.842076439112641446992775596814, 1.52415091243883269658405109505, 2.85737428242113121084998063579, 3.56751992307990314646414461845, 4.09760919714430160522336826497, 5.88635703820719160825427441810, 6.50105981175222189511746788803, 6.93761789977336294657755676560, 7.55150283510481960556303436374, 8.347162052350956582743007431620

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