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Properties

Label 2-3645-135.59-c0-0-1
Degree 22
Conductor 36453645
Sign −0.957−0.286i-0.957 - 0.286i
Analytic cond. 1.819091.81909
Root an. cond. 1.348731.34873
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  + (−0.766 + 0.642i)2-s + (−0.939 + 0.342i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (−0.939 − 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (0.347 + 1.96i)47-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)2-s + (−0.939 + 0.342i)5-s + (−0.499 − 0.866i)8-s + (0.5 − 0.866i)10-s + (0.939 + 0.342i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.173 + 0.984i)23-s + (0.766 − 0.642i)25-s + (−0.173 + 0.984i)31-s + (0.173 + 0.984i)34-s + (−0.939 − 0.342i)38-s + (0.766 + 0.642i)40-s + (−0.5 − 0.866i)46-s + (0.347 + 1.96i)47-s + ⋯

Functional equation

Λ(s)=(3645s/2 ΓC(s) L(s)=((−0.957−0.286i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3645s/2 ΓC(s) L(s)=((−0.957−0.286i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36453645    =    36⋅53^{6} \cdot 5
Sign: −0.957−0.286i-0.957 - 0.286i
Analytic conductor: 1.819091.81909
Root analytic conductor: 1.348731.34873
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3645(404,⋅)\chi_{3645} (404, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3645, ( :0), −0.957−0.286i)(2,\ 3645,\ (\ :0),\ -0.957 - 0.286i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.45259346570.4525934657
L(12)L(\frac12) ≈\approx 0.45259346570.4525934657
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+(0.939−0.342i)T 1 + (0.939 - 0.342i)T
good2 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
7 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
11 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
13 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
17 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(−0.347−1.96i)T+(−0.939+0.342i)T2 1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
61 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2}
67 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
83 1+(0.766−0.642i)T+(0.173−0.984i)T2 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)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

−8.978845207606328541271787010768, −7.929964727361603878114096638102, −7.83633548712610345262543273185, −7.07243496524060882261879856842, −6.38145692455421572384444005106, −5.44873230197759633240136283813, −4.43340708462142710077882539718, −3.48699743339343711447598302716, −2.98791396666301419606232888925, −1.23515980323002493233666220642, 0.39039799677121155815876909475, 1.54330409116017070381441533477, 2.64299498112681813761917726903, 3.57099887046083271141538820372, 4.50076985377928656162454540485, 5.29465085499758873120679671052, 6.14451129927002733064659367475, 7.12881904652342990543213036543, 7.924267155224486419945892409552, 8.520796087162132686261864995829

Graph of the ZZ-function along the critical line