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Properties

Label 2-2523-87.71-c0-0-0
Degree $2$
Conductor $2523$
Sign $-0.314 - 0.949i$
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  + (−0.900 − 0.433i)2-s + (−0.623 + 0.781i)3-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.222 + 0.974i)11-s + (0.222 − 0.974i)13-s + (0.900 − 0.433i)14-s + (0.222 − 0.974i)16-s + 17-s + (−0.222 + 0.974i)18-s + (−0.222 − 0.974i)21-s + (0.623 − 0.781i)22-s + (−0.900 − 0.433i)24-s + (0.623 + 0.781i)25-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.623 + 0.781i)3-s + (0.900 − 0.433i)6-s + (−0.623 + 0.781i)7-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.222 + 0.974i)11-s + (0.222 − 0.974i)13-s + (0.900 − 0.433i)14-s + (0.222 − 0.974i)16-s + 17-s + (−0.222 + 0.974i)18-s + (−0.222 − 0.974i)21-s + (0.623 − 0.781i)22-s + (−0.900 − 0.433i)24-s + (0.623 + 0.781i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.314 - 0.949i$
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: \((2,\ 2523,\ (\ :0),\ -0.314 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3520970238\)
\(L(\frac12)\) \(\approx\) \(0.3520970238\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.540172785711098983116128635499, −8.832206987440601878964852466730, −8.092852216569701005183985699904, −7.11564349631859017519513342265, −6.04197611852888056829340575356, −5.36225789719897735464942901136, −4.83062721460604297012433272404, −3.50239047565153207995928018258, −2.66898881811801412377626916242, −1.25384111235549620804038858402, 0.39937731199647628735268385611, 1.50462094085126324272121986629, 3.11349121948867926026910832906, 3.99952375747715530141429941475, 5.13093807706305171046303654684, 6.15871004890436190954700413164, 6.78382218466690434407531593526, 7.22948390384916180989620825671, 8.215259273193081138749147548105, 8.528599312973173496520084197806

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