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Properties

Label 2-3536-884.763-c0-0-0
Degree $2$
Conductor $3536$
Sign $-0.821 - 0.569i$
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  + (−1.83 + 0.758i)5-s + (0.965 + 0.258i)9-s + (0.258 + 0.965i)13-s + (−0.866 − 0.5i)17-s + (2.07 − 2.07i)25-s + (−0.207 + 1.57i)29-s + (−0.258 − 0.0340i)37-s + (−1.25 + 0.965i)41-s + (−1.96 + 0.258i)45-s + (−0.965 + 0.258i)49-s + (−1.36 − 1.36i)53-s + (0.258 + 1.96i)61-s + (−1.20 − 1.57i)65-s + (−1.46 + 0.607i)73-s + (0.866 + 0.499i)81-s + ⋯
L(s)  = 1  + (−1.83 + 0.758i)5-s + (0.965 + 0.258i)9-s + (0.258 + 0.965i)13-s + (−0.866 − 0.5i)17-s + (2.07 − 2.07i)25-s + (−0.207 + 1.57i)29-s + (−0.258 − 0.0340i)37-s + (−1.25 + 0.965i)41-s + (−1.96 + 0.258i)45-s + (−0.965 + 0.258i)49-s + (−1.36 − 1.36i)53-s + (0.258 + 1.96i)61-s + (−1.20 − 1.57i)65-s + (−1.46 + 0.607i)73-s + (0.866 + 0.499i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $-0.821 - 0.569i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (1647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ -0.821 - 0.569i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5220527029\)
\(L(\frac12)\) \(\approx\) \(0.5220527029\)
\(L(1)\) 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 \( 1 + (1.83 - 0.758i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.207 - 1.57i)T + (-0.965 - 0.258i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.258 + 0.0340i)T + (0.965 + 0.258i)T^{2} \)
41 \( 1 + (1.25 - 0.965i)T + (0.258 - 0.965i)T^{2} \)
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 \( 1 + (-0.258 - 1.96i)T + (-0.965 + 0.258i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.258 + 0.965i)T^{2} \)
73 \( 1 + (1.46 - 0.607i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.607 + 0.465i)T + (0.258 + 0.965i)T^{2} \)
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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

−8.808074964331946621359866429064, −8.298467300993411030821624939354, −7.35632968136373675748184608140, −7.01253559634900717466198656282, −6.45497062147476526585466393958, −4.91119743745085838612757978913, −4.40251186818376607782488942807, −3.66441211681263482300147150450, −2.88805191122178220512978950977, −1.55166390943379865575147470423, 0.32091265973326575317320311188, 1.58613962127286739245430600624, 3.15246920456414360269179410995, 3.91595694011988069654178528457, 4.44048024803992991333713283246, 5.20184036735225711817852603188, 6.34310838171366686873626433548, 7.14311957848969143301061835754, 7.86864411253276773592448631948, 8.256977216062574306828647689160

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