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Properties

Label 2-3528-504.131-c0-0-4
Degree $2$
Conductor $3528$
Sign $-0.0472 - 0.998i$
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  + i·2-s + (0.991 + 0.130i)3-s − 4-s + (−0.130 + 0.991i)6-s − i·8-s + (0.965 + 0.258i)9-s + (−0.448 − 0.258i)11-s + (−0.991 − 0.130i)12-s + 16-s + (0.130 + 0.226i)17-s + (−0.258 + 0.965i)18-s + (1.05 + 0.608i)19-s + (0.258 − 0.448i)22-s + (0.130 − 0.991i)24-s + (−0.5 + 0.866i)25-s + ⋯
L(s)  = 1  + i·2-s + (0.991 + 0.130i)3-s − 4-s + (−0.130 + 0.991i)6-s − i·8-s + (0.965 + 0.258i)9-s + (−0.448 − 0.258i)11-s + (−0.991 − 0.130i)12-s + 16-s + (0.130 + 0.226i)17-s + (−0.258 + 0.965i)18-s + (1.05 + 0.608i)19-s + (0.258 − 0.448i)22-s + (0.130 − 0.991i)24-s + (−0.5 + 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0472 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0472 - 0.998i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.0472 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.663918748\)
\(L(\frac12)\) \(\approx\) \(1.663918748\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-0.991 - 0.130i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.448 + 0.258i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.130 - 0.226i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1.05 - 0.608i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.991 + 1.71i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - 1.58T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.73T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.37 - 0.793i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.923 - 1.60i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.382 + 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-1.71 + 0.991i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.881872674976468873235280462153, −7.997748347442535274659761782441, −7.62419280321426479057130207023, −6.96760720634045034429281886669, −5.87982950972626888507805113632, −5.32398870196148271373384609862, −4.28928933754334706007420149446, −3.63534334486111762838843820997, −2.73863309127722130825437226497, −1.33000871933692912725118811023, 1.03525561396293075791187793925, 2.23153992826095682723948355867, 2.81909281068017641248656149604, 3.68968668618179487788727938008, 4.48793620871826525194611418419, 5.24450657078353387720870410068, 6.32657177691467461871599066888, 7.56405866471410351947656666390, 7.77141733928843190406487109958, 8.898467812764384140229972089970

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