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Properties

Label 2-3528-168.131-c0-0-0
Degree $2$
Conductor $3528$
Sign $-0.405 - 0.914i$
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  + (−0.258 − 0.965i)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 + (1 + 0.999i)22-s + (−0.707 + 1.22i)23-s + (−0.5 − 0.866i)25-s − 1.41·29-s + (−0.965 − 0.258i)32-s + (−1.73 − i)37-s + (0.707 − 1.22i)44-s + (1.36 + 0.366i)46-s + (−0.707 + 0.707i)50-s + (−0.707 − 1.22i)53-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)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 + (1 + 0.999i)22-s + (−0.707 + 1.22i)23-s + (−0.5 − 0.866i)25-s − 1.41·29-s + (−0.965 − 0.258i)32-s + (−1.73 − i)37-s + (0.707 − 1.22i)44-s + (1.36 + 0.366i)46-s + (−0.707 + 0.707i)50-s + (−0.707 − 1.22i)53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.405 - 0.914i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.405 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1131025345\)
\(L(\frac12)\) \(\approx\) \(0.1131025345\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
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

−9.153246410293632410566568159584, −8.271064660624387123923054650072, −7.69787018187870911564938471686, −7.04546056968344386393238530180, −5.66382798646927626819940975365, −5.17584554472534923750116087012, −4.16292213489820264935238902346, −3.46883528625518137505445595394, −2.38614415516310752368940977267, −1.72795365529004449695429138533, 0.06837937282000647650238937318, 1.72432674491784480502389804280, 3.05552205755259613898249554663, 4.01122675619973942491815886913, 4.99060853986606109168887191656, 5.57353560581480851815006888905, 6.26933471810995962568391421931, 7.11227894207126052096738218755, 7.84686071608115768448706681655, 8.326545082606151271369983132312

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