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Properties

Label 2-3536-884.399-c0-0-0
Degree $2$
Conductor $3536$
Sign $-0.0694 - 0.997i$
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 + (−0.258 + 0.965i)9-s + (−0.258 − 0.965i)13-s − i·17-s + (2.07 + 2.07i)25-s + (−1.25 + 0.965i)29-s + (0.158 + 0.207i)37-s + (−0.207 + 1.57i)41-s + (1.20 − 1.57i)45-s + (0.258 + 0.965i)49-s + (−1.36 + 1.36i)53-s + (1.57 + 1.20i)61-s + (−0.258 + 1.96i)65-s + (−1.46 − 0.607i)73-s + (−0.866 − 0.499i)81-s + ⋯
L(s)  = 1  + (−1.83 − 0.758i)5-s + (−0.258 + 0.965i)9-s + (−0.258 − 0.965i)13-s − i·17-s + (2.07 + 2.07i)25-s + (−1.25 + 0.965i)29-s + (0.158 + 0.207i)37-s + (−0.207 + 1.57i)41-s + (1.20 − 1.57i)45-s + (0.258 + 0.965i)49-s + (−1.36 + 1.36i)53-s + (1.57 + 1.20i)61-s + (−0.258 + 1.96i)65-s + (−1.46 − 0.607i)73-s + (−0.866 − 0.499i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0694 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $-0.0694 - 0.997i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (399, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ -0.0694 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4075634960\)
\(L(\frac12)\) \(\approx\) \(0.4075634960\)
\(L(1)\) 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 \( 1 + iT \)
good3 \( 1 + (0.258 - 0.965i)T^{2} \)
5 \( 1 + (1.83 + 0.758i)T + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (-0.258 - 0.965i)T^{2} \)
11 \( 1 + (-0.965 - 0.258i)T^{2} \)
19 \( 1 + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.965 - 0.258i)T^{2} \)
29 \( 1 + (1.25 - 0.965i)T + (0.258 - 0.965i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.158 - 0.207i)T + (-0.258 + 0.965i)T^{2} \)
41 \( 1 + (0.207 - 1.57i)T + (-0.965 - 0.258i)T^{2} \)
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 \( 1 + (-1.57 - 1.20i)T + (0.258 + 0.965i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.965 + 0.258i)T^{2} \)
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 \( 1 + (0.0999 + 0.758i)T + (-0.965 + 0.258i)T^{2} \)
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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

−8.773799108203934409913713275798, −8.042979284552620624287803696582, −7.62718784378169237143926659965, −7.09883980206148693204257332169, −5.73054085286132064795498077853, −4.94023004604747352113017668162, −4.51533491907903141301217025955, −3.46098726582716841106992669200, −2.76261172933659029915187055230, −1.15212896993315467230500717658, 0.26993801071272039359165167898, 2.07267781265642578310200451326, 3.31150733472183035030016546397, 3.83604199787752130013235643880, 4.36977123821304361447137530270, 5.62381808766045704049084884216, 6.65404610550021349288138391922, 6.96664465956362825837469157704, 7.85349265157187393475995695941, 8.407128613033974930014814358523

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