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Properties

Label 2-3528-8.3-c0-0-0
Degree $2$
Conductor $3528$
Sign $-i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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 + 16-s + 2i·23-s + 25-s − 2i·29-s + i·32-s + 2·43-s − 2·46-s + i·50-s + 2i·53-s + 2·58-s − 64-s + 2·67-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s + 16-s + 2i·23-s + 25-s − 2i·29-s + i·32-s + 2·43-s − 2·46-s + i·50-s + 2i·53-s + 2·58-s − 64-s + 2·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120770555\)
\(L(\frac12)\) \(\approx\) \(1.120770555\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 2T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.899361819602058303556580214097, −7.964367688141004545372412970588, −7.52776675909715434508590690277, −6.75589416357485867414871967653, −5.88824008691254355784721637350, −5.41520702284285638226380296923, −4.39743715455457080207742659710, −3.76621642528939160337731498578, −2.61576119687548318090402244022, −1.08731138611794378212789298646, 0.843981798799303935556929761975, 2.08613502998531756224438738549, 2.91982300187330564139886741939, 3.76421719505902442970045466875, 4.69687866381282804432004504054, 5.22450797416426672194843837321, 6.31777272154704781262586250683, 7.08637248517300635981022897677, 8.124673474241451557566521976661, 8.725541625605721927706512258917

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