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Properties

Label 2-2523-87.71-c0-0-4
Degree $2$
Conductor $2523$
Sign $0.942 - 0.335i$
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  + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.385 − 0.483i)7-s + (0.222 + 0.974i)9-s − 0.999i·12-s + (−0.360 + 1.57i)13-s + (−0.222 + 0.974i)16-s + (1.26 − 1.00i)19-s + (0.602 − 0.137i)21-s + (0.623 + 0.781i)25-s + (−0.433 + 0.900i)27-s − 0.618·28-s + (0.268 − 0.556i)31-s + (0.623 − 0.781i)36-s + (1.57 − 0.360i)37-s + ⋯
L(s)  = 1  + (0.781 + 0.623i)3-s + (−0.623 − 0.781i)4-s + (0.385 − 0.483i)7-s + (0.222 + 0.974i)9-s − 0.999i·12-s + (−0.360 + 1.57i)13-s + (−0.222 + 0.974i)16-s + (1.26 − 1.00i)19-s + (0.602 − 0.137i)21-s + (0.623 + 0.781i)25-s + (−0.433 + 0.900i)27-s − 0.618·28-s + (0.268 − 0.556i)31-s + (0.623 − 0.781i)36-s + (1.57 − 0.360i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.942 - 0.335i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (2333, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455110946\)
\(L(\frac12)\) \(\approx\) \(1.455110946\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.781 - 0.623i)T \)
29 \( 1 \)
good2 \( 1 + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.385 + 0.483i)T + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (0.360 - 1.57i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-1.26 + 1.00i)T + (0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.268 + 0.556i)T + (-0.623 - 0.781i)T^{2} \)
37 \( 1 + (-1.57 + 0.360i)T + (0.900 - 0.433i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.483 - 0.385i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 + (0.360 + 1.57i)T + (-0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.702 + 1.45i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 + (1.57 - 0.360i)T + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.483 + 0.385i)T + (0.222 - 0.974i)T^{2} \)
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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

−9.375219800465892533539719422857, −8.645276795649274092509916741694, −7.62181512236734829941441048950, −7.02425549273816468371628050513, −5.89943509515140031562867096852, −4.81162132200099413599885244953, −4.57363988868802748867685518966, −3.62198921196768281771707073153, −2.43033800641773447430458121297, −1.31644057167563678991335379583, 1.10318329860255211215828082534, 2.65441282000869974593491744312, 3.10246561225103306961909115897, 4.10365560758278354084279566051, 5.12759190305296098999319576364, 5.90945381171841174241666995994, 7.07619858775593197315347560630, 7.77197833128690329917996854020, 8.239484908283213971399584377041, 8.744556847472592540370265420983

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