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Properties

Label 2-2523-3.2-c0-0-0
Degree $2$
Conductor $2523$
Sign $-i$
Analytic cond. $1.25914$
Root an. cond. $1.12211$
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 − i·3-s + 6-s − 7-s + i·8-s − 9-s + i·11-s + 13-s − i·14-s − 16-s + i·17-s − i·18-s + i·21-s − 22-s + 24-s + 25-s + ⋯
L(s)  = 1  + i·2-s − i·3-s + 6-s − 7-s + i·8-s − 9-s + i·11-s + 13-s − i·14-s − 16-s + i·17-s − i·18-s + i·21-s − 22-s + 24-s + 25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (842, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.126783645\)
\(L(\frac12)\) \(\approx\) \(1.126783645\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
29 \( 1 \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + T + T^{2} \)
11 \( 1 - iT - T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - iT - T^{2} \)
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

−8.914647264808119551147033118802, −8.302330534995812690361097935032, −7.63289350754837642783939638484, −6.75776201656374617482564800148, −6.48837348102456735980562929759, −5.81783706053121053892286880332, −4.85560986587378876305451612939, −3.54795345643235749065271537539, −2.55393028917965622868668693700, −1.49383854093713695742919085709, 0.76082110537880551539079789685, 2.45728836171047692851071837289, 3.31629230445045059492151244925, 3.61433403540286288046286177507, 4.72496452624142138876593834835, 5.80858522734923972651757789858, 6.40208766299634909919109050041, 7.32972770064118580662439226818, 8.646948265710912958429567992212, 9.046429866415798144627632653152

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