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Properties

Label 2-3528-168.59-c0-0-1
Degree $2$
Conductor $3528$
Sign $0.914 + 0.405i$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (−1.22 − 0.707i)11-s + (0.500 + 0.866i)16-s + (0.999 + i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (−0.258 − 0.965i)32-s + (1.73 − i)37-s + (−0.707 − 1.22i)44-s + (−0.366 − 1.36i)46-s + (0.707 − 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.707 − 0.707i)8-s + (−1.22 − 0.707i)11-s + (0.500 + 0.866i)16-s + (0.999 + i)22-s + (0.707 + 1.22i)23-s + (−0.5 + 0.866i)25-s + 1.41·29-s + (−0.258 − 0.965i)32-s + (1.73 − i)37-s + (−0.707 − 1.22i)44-s + (−0.366 − 1.36i)46-s + (0.707 − 0.707i)50-s + (0.707 − 1.22i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.914 + 0.405i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.914 + 0.405i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7537335739\)
\(L(\frac12)\) \(\approx\) \(0.7537335739\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.682381917468510753125745933729, −7.970736788097982290236420100615, −7.52847520258263196176481585529, −6.63677961318306715547023793356, −5.79133545683087927219618318688, −5.05452067541768032546500491939, −3.73620790976095700264468687548, −2.98589715333756599241069348626, −2.12752316644489962314444327526, −0.823239394549130815403278841678, 0.908463878619858256183076698821, 2.37038297846789174328206373223, 2.79066412955536811545513521573, 4.37616833819932999060556296057, 5.08316626079114122221429638644, 6.03879332750755548456038818563, 6.70408432241950939820858426182, 7.44489035779570947389239096878, 8.186200661286028410879952625935, 8.576894926304923065699275308817

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