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Properties

Label 2-2523-87.5-c0-0-1
Degree $2$
Conductor $2523$
Sign $0.768 - 0.639i$
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  + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s + i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (0.602 + 0.137i)19-s + (−0.702 − 1.45i)21-s + (−0.222 + 0.974i)25-s + (−0.781 + 0.623i)27-s + 1.61·28-s + (−1.26 + 1.00i)31-s + (−0.222 − 0.974i)36-s + (−0.268 − 0.556i)37-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)3-s + (0.222 − 0.974i)4-s + (0.360 + 1.57i)7-s + (0.900 − 0.433i)9-s + i·12-s + (0.556 + 0.268i)13-s + (−0.900 − 0.433i)16-s + (0.602 + 0.137i)19-s + (−0.702 − 1.45i)21-s + (−0.222 + 0.974i)25-s + (−0.781 + 0.623i)27-s + 1.61·28-s + (−1.26 + 1.00i)31-s + (−0.222 − 0.974i)36-s + (−0.268 − 0.556i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2523\)    =    \(3 \cdot 29^{2}\)
Sign: $0.768 - 0.639i$
Analytic conductor: \(1.25914\)
Root analytic conductor: \(1.12211\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2523} (1745, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2523,\ (\ :0),\ 0.768 - 0.639i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9479089110\)
\(L(\frac12)\) \(\approx\) \(0.9479089110\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.974 - 0.222i)T \)
29 \( 1 \)
good2 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (-0.360 - 1.57i)T + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.556 - 0.268i)T + (0.623 + 0.781i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.602 - 0.137i)T + (0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (1.26 - 1.00i)T + (0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.268 + 0.556i)T + (-0.623 + 0.781i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1.26 - 1.00i)T + (0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.57 + 0.360i)T + (0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.556 + 0.268i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.483 - 0.385i)T + (0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.268 - 0.556i)T + (-0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 + 0.433i)T^{2} \)
89 \( 1 + (-0.222 + 0.974i)T^{2} \)
97 \( 1 + (-1.57 - 0.360i)T + (0.900 + 0.433i)T^{2} \)
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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

−9.317256861073392824076246013254, −8.677429515708931493482146371701, −7.46787430133799321981060530066, −6.63440995374880157456025891907, −5.82828481389231086585531353608, −5.45512431409651125413961067415, −4.82799088096735540646352264618, −3.58886033954051034692267720033, −2.21223559519177904096738983261, −1.31016054440299041107434375314, 0.800422261972713066664649704950, 2.13278273125476155769238280725, 3.62442701967029038241153605060, 4.11333105819360385564572359716, 5.00650714893477179702905186032, 6.04860350983906191130208908789, 6.85740551826042833432315854811, 7.48672024357924725841629784740, 7.88400497987356771135902019543, 8.899165881763060334203911383374

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