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Properties

Label 2-3536-884.679-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: $\chi_{3536} (2447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ 0.967 + 0.252i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.354043727\)
\(L(\frac12)\) \(\approx\) \(1.354043727\)
\(L(1)\) 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

−8.660380382186033829183614448154, −8.298235845594229578986418065882, −7.14528282249597440172384082838, −6.43732429486305027723759989600, −5.81313063804161829059750871730, −4.76140171461352614892125110792, −4.14696847155392410009254929592, −3.05421894165114264349556428055, −2.55441484153235566584712690370, −0.877364019924311599585794920524, 1.26805348888795297857244933991, 2.25525950314456447878398194755, 3.16194874096657726517613428712, 4.17504115153596792397631641680, 4.99839784667396684402596764484, 5.74431871917986791424178268944, 6.73991155349722987541758345471, 7.46974346560494339932097667620, 7.88170960502985967429050231679, 8.515716407105602714700652651115

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