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Properties

Label 2-3536-884.815-c0-0-2
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  + (−0.258 + 0.448i)3-s + (0.258 + 0.448i)7-s + (0.366 + 0.633i)9-s + (0.707 − 1.22i)11-s − i·13-s + (0.5 + 0.866i)17-s − 0.267·21-s + (0.965 − 1.67i)23-s + 25-s − 0.896·27-s − 1.93·31-s + (0.366 + 0.633i)33-s + (0.448 + 0.258i)39-s + (0.366 − 0.633i)49-s − 0.517·51-s + ⋯
L(s)  = 1  + (−0.258 + 0.448i)3-s + (0.258 + 0.448i)7-s + (0.366 + 0.633i)9-s + (0.707 − 1.22i)11-s − i·13-s + (0.5 + 0.866i)17-s − 0.267·21-s + (0.965 − 1.67i)23-s + 25-s − 0.896·27-s − 1.93·31-s + (0.366 + 0.633i)33-s + (0.448 + 0.258i)39-s + (0.366 − 0.633i)49-s − 0.517·51-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: $\chi_{3536} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.967 - 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.339735381\)
\(L(\frac12)\) \(\approx\) \(1.339735381\)
\(L(1)\) 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 \( 1 + (0.258 - 0.448i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + 1.93T + T^{2} \)
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 \( 1 - 0.517T + T^{2} \)
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

−8.613602985312947534322906812325, −8.306111208058616015661810967672, −7.29642945555616843654835997417, −6.47120367649517801998456499918, −5.54994501368582755708904476051, −5.20540403347211517287856685433, −4.12433581424030895938360418701, −3.36899203946352578079085497891, −2.35791714949294596290066820759, −1.04361934414661218097555840348, 1.18491289950744821566724559782, 1.89101392079146728649828930544, 3.32389834620789349705977441324, 4.10631572760992113838335360566, 4.89047813330372744512758869728, 5.73372321976078538433905813349, 6.85999704193364438461871261660, 7.10842216203936692486368625921, 7.60401896551007098686392155762, 9.026046974121180662741429532921

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