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Properties

Label 2-15e3-5.2-c0-0-2
Degree 22
Conductor 33753375
Sign −1-1
Analytic cond. 1.684341.68434
Root an. cond. 1.297821.29782
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  + (1.40 + 1.40i)2-s + 2.95i·4-s + (−2.75 + 2.75i)8-s − 4.78·16-s + (1.05 + 1.05i)17-s − 0.209i·19-s + (−0.294 + 0.294i)23-s − 1.33·31-s + (−3.97 − 3.97i)32-s + 2.95i·34-s + (0.294 − 0.294i)38-s − 0.827·46-s + (0.831 + 0.831i)47-s − i·49-s + (0.575 − 0.575i)53-s + ⋯
L(s)  = 1  + (1.40 + 1.40i)2-s + 2.95i·4-s + (−2.75 + 2.75i)8-s − 4.78·16-s + (1.05 + 1.05i)17-s − 0.209i·19-s + (−0.294 + 0.294i)23-s − 1.33·31-s + (−3.97 − 3.97i)32-s + 2.95i·34-s + (0.294 − 0.294i)38-s − 0.827·46-s + (0.831 + 0.831i)47-s − i·49-s + (0.575 − 0.575i)53-s + ⋯

Functional equation

Λ(s)=(3375s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3375s/2 ΓC(s) L(s)=(− Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33753375    =    33⋅533^{3} \cdot 5^{3}
Sign: −1-1
Analytic conductor: 1.684341.68434
Root analytic conductor: 1.297821.29782
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3375(1432,⋅)\chi_{3375} (1432, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3375, ( :0), −1)(2,\ 3375,\ (\ :0),\ -1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.4448706602.444870660
L(12)L(\frac12) ≈\approx 2.4448706602.444870660
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)
bad3 1 1
5 1 1
good2 1+(−1.40−1.40i)T+iT2 1 + (-1.40 - 1.40i)T + iT^{2}
7 1+iT2 1 + iT^{2}
11 1+T2 1 + T^{2}
13 1−iT2 1 - iT^{2}
17 1+(−1.05−1.05i)T+iT2 1 + (-1.05 - 1.05i)T + iT^{2}
19 1+0.209iT−T2 1 + 0.209iT - T^{2}
23 1+(0.294−0.294i)T−iT2 1 + (0.294 - 0.294i)T - iT^{2}
29 1−T2 1 - T^{2}
31 1+1.33T+T2 1 + 1.33T + T^{2}
37 1+iT2 1 + iT^{2}
41 1+T2 1 + T^{2}
43 1−iT2 1 - iT^{2}
47 1+(−0.831−0.831i)T+iT2 1 + (-0.831 - 0.831i)T + iT^{2}
53 1+(−0.575+0.575i)T−iT2 1 + (-0.575 + 0.575i)T - iT^{2}
59 1−T2 1 - T^{2}
61 1−1.82T+T2 1 - 1.82T + T^{2}
67 1+iT2 1 + iT^{2}
71 1+T2 1 + T^{2}
73 1−iT2 1 - iT^{2}
79 1+1.82iT−T2 1 + 1.82iT - T^{2}
83 1+(0.575−0.575i)T−iT2 1 + (0.575 - 0.575i)T - iT^{2}
89 1−T2 1 - T^{2}
97 1+iT2 1 + iT^{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

−8.707248398435790259416492649657, −8.167382709151010676479120769124, −7.42190366085116364974357928470, −6.89574925692158753917385695927, −5.92064490874512054635908200962, −5.59612685501772520913413410014, −4.72479917295082360887864063188, −3.83177943084697839795209948799, −3.35296993191824917866157549070, −2.15468120209287805675107230215, 0.922265858554594501490798561337, 2.05259722822569922707255922239, 2.87889391208419212531473712069, 3.67039222024123940723395906997, 4.35881016260369444063880672141, 5.34453251766047044615205563134, 5.64041006078442877422895961344, 6.66171079726048173470960738635, 7.45091301345448680029434584173, 8.767623984785994925017958332421

Graph of the ZZ-function along the critical line