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Properties

Label 2-2523-87.35-c0-0-5
Degree $2$
Conductor $2523$
Sign $-0.510 + 0.859i$
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  + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)14-s + (0.900 − 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (−0.900 − 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (0.222 − 0.974i)3-s + (−0.623 − 0.781i)6-s + (0.222 − 0.974i)7-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.900 + 0.433i)11-s + (0.900 − 0.433i)13-s + (−0.623 − 0.781i)14-s + (0.900 − 0.433i)16-s + 17-s + (−0.900 + 0.433i)18-s + (−0.900 − 0.433i)21-s + (−0.222 + 0.974i)22-s + (0.623 − 0.781i)24-s + (−0.222 − 0.974i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.510 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $-0.510 + 0.859i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ -0.510 + 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.922582196\)
\(L(\frac12)\) \(\approx\) \(1.922582196\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.476916428660894949141321251000, −8.004983950815997236101789991679, −7.45062501746509790970548503566, −6.63921139502084078521776783551, −5.60146544357262729571112952604, −4.76503272919922634093086083256, −3.71849081307201234511565816427, −3.09780613640425390563840335533, −2.09428098455190613825654761652, −1.11613387356117617558569275510, 1.78415266742459309156314613085, 3.10479174910509789356280881722, 3.82716054622805759268503621344, 4.94915848960930964941216748287, 5.40751521242256998651233432627, 5.92236518718995601478334237267, 6.87988696960492054353713204678, 8.001730339250659605469497004542, 8.410449471510493092654716841845, 9.325040114977261236496363492783

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