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Properties

Label 2-3536-3536.2413-c0-0-0
Degree $2$
Conductor $3536$
Sign $-0.0985 - 0.995i$
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  + i·2-s − 4-s − i·8-s + i·9-s + 13-s + 16-s − i·17-s − 18-s + 25-s + i·26-s + i·32-s + 34-s − i·36-s + (1 + i)43-s + (−1 + i)47-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + i·9-s + 13-s + 16-s − i·17-s − 18-s + 25-s + i·26-s + i·32-s + 34-s − i·36-s + (1 + i)43-s + (−1 + i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3536\)    =    \(2^{4} \cdot 13 \cdot 17\)
Sign: $-0.0985 - 0.995i$
Analytic conductor: \(1.76469\)
Root analytic conductor: \(1.32841\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3536} (2413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3536,\ (\ :0),\ -0.0985 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.185480214\)
\(L(\frac12)\) \(\approx\) \(1.185480214\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
13 \( 1 - T \)
17 \( 1 + iT \)
good3 \( 1 - iT^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (-1 - i)T + iT^{2} \)
47 \( 1 + (1 - i)T - iT^{2} \)
53 \( 1 + (-1 + i)T - iT^{2} \)
59 \( 1 + 2iT - T^{2} \)
61 \( 1 - iT^{2} \)
67 \( 1 - 2iT - T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 - i)T + iT^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.774928006619731586509522327119, −8.091368099698612157353936631787, −7.51243779814245828348534132376, −6.72305446123257939917405861857, −6.04029249044264532633852902845, −5.15770523115241402063673278386, −4.65968767527883328381940452275, −3.66866879078912796917283668085, −2.63723080338795223584587636732, −1.14245615612090858316260103397, 0.890975894662887887774005334570, 1.90763261359856283758483642896, 3.10109777706804962146086309245, 3.73683308362411777072965207867, 4.42206552374936721207523348102, 5.51904683279021206779188879020, 6.16291561746106680090393365224, 7.05282987940095833570998570000, 8.126785444963654914310774699654, 8.829437303798602176396520639121

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