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Properties

Label 2-3528-392.61-c0-0-0
Degree $2$
Conductor $3528$
Sign $0.138 + 0.990i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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  + (−0.563 − 0.826i)2-s + (−0.365 + 0.930i)4-s + (−1.90 − 0.587i)5-s + (0.955 − 0.294i)7-s + (0.974 − 0.222i)8-s + (0.587 + 1.90i)10-s + (0.728 − 0.0546i)11-s + (−0.781 − 0.623i)14-s + (−0.733 − 0.680i)16-s + (1.24 − 1.55i)20-s + (−0.455 − 0.571i)22-s + (2.46 + 1.67i)25-s + (−0.0747 + 0.997i)28-s + (1.29 + 1.03i)29-s + (−1.17 + 0.680i)31-s + (−0.149 + 0.988i)32-s + ⋯
L(s)  = 1  + (−0.563 − 0.826i)2-s + (−0.365 + 0.930i)4-s + (−1.90 − 0.587i)5-s + (0.955 − 0.294i)7-s + (0.974 − 0.222i)8-s + (0.587 + 1.90i)10-s + (0.728 − 0.0546i)11-s + (−0.781 − 0.623i)14-s + (−0.733 − 0.680i)16-s + (1.24 − 1.55i)20-s + (−0.455 − 0.571i)22-s + (2.46 + 1.67i)25-s + (−0.0747 + 0.997i)28-s + (1.29 + 1.03i)29-s + (−1.17 + 0.680i)31-s + (−0.149 + 0.988i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.138 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.138 + 0.990i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.138 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7181570099\)
\(L(\frac12)\) \(\approx\) \(0.7181570099\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.563 + 0.826i)T \)
3 \( 1 \)
7 \( 1 + (-0.955 + 0.294i)T \)
good5 \( 1 + (1.90 + 0.587i)T + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.728 + 0.0546i)T + (0.988 - 0.149i)T^{2} \)
13 \( 1 + (0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 - 0.294i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (-1.29 - 1.03i)T + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (1.17 - 0.680i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.930 + 0.365i)T + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (-1.65 + 0.510i)T + (0.826 - 0.563i)T^{2} \)
61 \( 1 + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-1.09 + 1.61i)T + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.955 + 1.65i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (1.01 - 0.488i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
97 \( 1 - 0.298iT - T^{2} \)
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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

−8.709870666440554789950290768188, −7.986069590089053292259347785053, −7.42970545359024913630854482158, −6.77808329345759106485850279686, −5.02929439331236098428168648698, −4.60722364757562735046082833705, −3.76737958241484875833580287594, −3.25346932678602486229082540121, −1.70873067938099022490237462227, −0.76190288290450746138197225704, 0.921186310074318889255220034029, 2.39917787604119866447913348119, 3.77865593522100051037070889865, 4.33041302834479922714081344349, 5.10839093050537664349173399532, 6.18054258711383550488360344109, 6.95148358496379026772488391818, 7.50723438918421965063499122683, 8.180512675633506202424604002304, 8.515572271214408092138841066282

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