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Properties

Label 2-3536-884.87-c0-0-0
Degree $2$
Conductor $3536$
Sign $-0.668 - 0.743i$
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  + (0.607 + 1.46i)5-s + (−0.965 − 0.258i)9-s + (−0.965 + 0.258i)13-s + i·17-s + (−1.07 + 1.07i)25-s + (−1.96 − 0.258i)29-s + (−0.158 + 1.20i)37-s + (1.20 + 1.57i)41-s + (−0.207 − 1.57i)45-s + (0.965 − 0.258i)49-s + (0.366 + 0.366i)53-s + (−1.57 + 0.207i)61-s + (−0.965 − 1.25i)65-s + (−0.758 − 1.83i)73-s + (0.866 + 0.499i)81-s + ⋯
L(s)  = 1  + (0.607 + 1.46i)5-s + (−0.965 − 0.258i)9-s + (−0.965 + 0.258i)13-s + i·17-s + (−1.07 + 1.07i)25-s + (−1.96 − 0.258i)29-s + (−0.158 + 1.20i)37-s + (1.20 + 1.57i)41-s + (−0.207 − 1.57i)45-s + (0.965 − 0.258i)49-s + (0.366 + 0.366i)53-s + (−1.57 + 0.207i)61-s + (−0.965 − 1.25i)65-s + (−0.758 − 1.83i)73-s + (0.866 + 0.499i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $-0.668 - 0.743i$
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: \((2,\ 3536,\ (\ :0),\ -0.668 - 0.743i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9060549456\)
\(L(\frac12)\) \(\approx\) \(0.9060549456\)
\(L(1)\) 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 \( 1 + (-0.607 - 1.46i)T + (-0.707 + 0.707i)T^{2} \)
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 \( 1 + (1.96 + 0.258i)T + (0.965 + 0.258i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.158 - 1.20i)T + (-0.965 - 0.258i)T^{2} \)
41 \( 1 + (-1.20 - 1.57i)T + (-0.258 + 0.965i)T^{2} \)
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 \( 1 + (1.57 - 0.207i)T + (0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.258 - 0.965i)T^{2} \)
73 \( 1 + (0.758 + 1.83i)T + (-0.707 + 0.707i)T^{2} \)
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

−9.268552702940511807712184397502, −8.141750232759394913033172384607, −7.49883956601159606078654773420, −6.70671665626055361834611514685, −6.06869008345437769200888953764, −5.53916578533286375377163361202, −4.34670233201534384459144231837, −3.32489814753124103396545118388, −2.68573255567198705820496670675, −1.83742370597555535591924669022, 0.48314063890882040608677459928, 1.90473692936158133708360767864, 2.68297601521050924238559347877, 3.93952298000857555475753027579, 4.86397281291433775658289653290, 5.51847965243651092643284202219, 5.80599102706744437093503292658, 7.23636022359606667976318190956, 7.68947458544313478646437984340, 8.761992876610385078266930168093

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