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Properties

Label 2-3528-168.131-c0-0-0
Degree 22
Conductor 35283528
Sign −0.405−0.914i-0.405 - 0.914i
Analytic cond. 1.760701.76070
Root an. cond. 1.326911.32691
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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

Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.405−0.914i) Λ‾(1−s)\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}
Λ(s)=(3528s/2 ΓC(s) L(s)=((−0.405−0.914i) Λ‾(1−s)\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: 22
Conductor: 35283528    =    23⋅32⋅722^{3} \cdot 3^{2} \cdot 7^{2}
Sign: −0.405−0.914i-0.405 - 0.914i
Analytic conductor: 1.760701.76070
Root analytic conductor: 1.326911.32691
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3528(1979,⋅)\chi_{3528} (1979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3528, ( :0), −0.405−0.914i)(2,\ 3528,\ (\ :0),\ -0.405 - 0.914i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.11310253450.1131025345
L(12)L(\frac12) ≈\approx 0.11310253450.1131025345
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.258+0.965i)T 1 + (0.258 + 0.965i)T
3 1 1
7 1 1
good5 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
11 1+(1.22−0.707i)T+(0.5−0.866i)T2 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2}
13 1+T2 1 + T^{2}
17 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(0.707−1.22i)T+(−0.5−0.866i)T2 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}
29 1+1.41T+T2 1 + 1.41T + T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+(1.73+i)T+(0.5+0.866i)T2 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2}
41 1+T2 1 + T^{2}
43 1+T2 1 + T^{2}
47 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
53 1+(0.707+1.22i)T+(−0.5+0.866i)T2 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}
59 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
67 1+(−1−1.73i)T+(−0.5+0.866i)T2 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2}
71 1+1.41T+T2 1 + 1.41T + T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1+T2 1 + T^{2}
89 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
97 1−T2 1 - T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line