↑

Properties

Label 2-3536-884.263-c0-0-0
Degree $2$
Conductor $3536$
Sign $0.996 - 0.0822i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.12 + 0.465i)5-s + (0.965 − 0.258i)9-s + (0.965 + 0.258i)13-s − i·17-s + (0.341 + 0.341i)25-s + (−0.0340 − 0.258i)29-s + (−1.57 + 0.207i)37-s + (−0.207 − 0.158i)41-s + (1.20 + 0.158i)45-s + (−0.965 − 0.258i)49-s + (0.366 − 0.366i)53-s + (−0.158 + 1.20i)61-s + (0.965 + 0.741i)65-s + (−0.241 − 0.0999i)73-s + (0.866 − 0.499i)81-s + ⋯
L(s)  = 1  + (1.12 + 0.465i)5-s + (0.965 − 0.258i)9-s + (0.965 + 0.258i)13-s − i·17-s + (0.341 + 0.341i)25-s + (−0.0340 − 0.258i)29-s + (−1.57 + 0.207i)37-s + (−0.207 − 0.158i)41-s + (1.20 + 0.158i)45-s + (−0.965 − 0.258i)49-s + (0.366 − 0.366i)53-s + (−0.158 + 1.20i)61-s + (0.965 + 0.741i)65-s + (−0.241 − 0.0999i)73-s + (0.866 − 0.499i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $0.996 - 0.0822i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (2031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.996 - 0.0822i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.760001208\)
\(L(\frac12)\) \(\approx\) \(1.760001208\)
\(L(1)\) 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 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + iT \)
good3 \( 1 + (-0.965 + 0.258i)T^{2} \)
5 \( 1 + (-1.12 - 0.465i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.0340 + 0.258i)T + (-0.965 + 0.258i)T^{2} \)
31 \( 1 + (0.707 + 0.707i)T^{2} \)
37 \( 1 + (1.57 - 0.207i)T + (0.965 - 0.258i)T^{2} \)
41 \( 1 + (0.207 + 0.158i)T + (0.258 + 0.965i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.366 + 0.366i)T - iT^{2} \)
59 \( 1 + (-0.866 + 0.5i)T^{2} \)
61 \( 1 + (0.158 - 1.20i)T + (-0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.258 - 0.965i)T^{2} \)
73 \( 1 + (0.241 + 0.0999i)T + (0.707 + 0.707i)T^{2} \)
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 \( 1 + (0.607 - 0.465i)T + (0.258 - 0.965i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.939843402556796987229186249494, −8.016615357229412424682881840735, −6.99609535244158228774667400838, −6.65852151591950983869880226916, −5.81892841171060366639611514960, −5.06567012922169522476874259440, −4.09275874150732023265950109633, −3.20974008903981208177965894576, −2.16969974153752866388163387404, −1.30451498561095344976216545250, 1.40212629615332874493893671279, 1.89887072382032970389084979477, 3.26901620750522454862907677715, 4.14958644170869149995526055349, 5.04111129773378743666338485371, 5.75521150984981049949142033098, 6.40789589797804151967206841234, 7.18740334768857673303147919485, 8.151635306959622309797002606851, 8.735301542244656622255818453225

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