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Properties

Label 2-15e2-3.2-c2-0-8
Degree 22
Conductor 225225
Sign −0.577+0.816i-0.577 + 0.816i
Analytic cond. 6.130806.13080
Root an. cond. 2.476042.47604
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.64i·2-s − 3.00·4-s + 11.2·7-s − 2.64i·8-s − 4.24i·11-s + 11.2·13-s − 29.6i·14-s − 18.9·16-s − 10.5i·17-s − 20·19-s − 11.2·22-s − 5.29i·23-s − 29.6i·26-s − 33.6·28-s − 8.48i·29-s + ⋯
L(s)  = 1  − 1.32i·2-s − 0.750·4-s + 1.60·7-s − 0.330i·8-s − 0.385i·11-s + 0.863·13-s − 2.12i·14-s − 1.18·16-s − 0.622i·17-s − 1.05·19-s − 0.510·22-s − 0.230i·23-s − 1.14i·26-s − 1.20·28-s − 0.292i·29-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.577+0.816i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1) L(s)=((−0.577+0.816i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.577+0.816i-0.577 + 0.816i
Analytic conductor: 6.130806.13080
Root analytic conductor: 2.476042.47604
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ225(26,⋅)\chi_{225} (26, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1), −0.577+0.816i)(2,\ 225,\ (\ :1),\ -0.577 + 0.816i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.837380−1.61769i0.837380 - 1.61769i
L(12)L(\frac12) ≈\approx 0.837380−1.61769i0.837380 - 1.61769i
L(2)L(2) 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+2.64iT−4T2 1 + 2.64iT - 4T^{2}
7 1−11.2T+49T2 1 - 11.2T + 49T^{2}
11 1+4.24iT−121T2 1 + 4.24iT - 121T^{2}
13 1−11.2T+169T2 1 - 11.2T + 169T^{2}
17 1+10.5iT−289T2 1 + 10.5iT - 289T^{2}
19 1+20T+361T2 1 + 20T + 361T^{2}
23 1+5.29iT−529T2 1 + 5.29iT - 529T^{2}
29 1+8.48iT−841T2 1 + 8.48iT - 841T^{2}
31 1−26T+961T2 1 - 26T + 961T^{2}
37 1+33.6T+1.36e3T2 1 + 33.6T + 1.36e3T^{2}
41 1−55.1iT−1.68e3T2 1 - 55.1iT - 1.68e3T^{2}
43 1−22.4T+1.84e3T2 1 - 22.4T + 1.84e3T^{2}
47 1−21.1iT−2.20e3T2 1 - 21.1iT - 2.20e3T^{2}
53 1+84.6iT−2.80e3T2 1 + 84.6iT - 2.80e3T^{2}
59 1−46.6iT−3.48e3T2 1 - 46.6iT - 3.48e3T^{2}
61 1+22T+3.72e3T2 1 + 22T + 3.72e3T^{2}
67 1−89.7T+4.48e3T2 1 - 89.7T + 4.48e3T^{2}
71 1−50.9iT−5.04e3T2 1 - 50.9iT - 5.04e3T^{2}
73 1−67.3T+5.32e3T2 1 - 67.3T + 5.32e3T^{2}
79 1+14T+6.24e3T2 1 + 14T + 6.24e3T^{2}
83 1−74.0iT−6.88e3T2 1 - 74.0iT - 6.88e3T^{2}
89 1−89.0iT−7.92e3T2 1 - 89.0iT - 7.92e3T^{2}
97 1+22.4T+9.40e3T2 1 + 22.4T + 9.40e3T^{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

−11.38020456201942924699480149328, −11.07251277493000026820162927007, −10.08898216862017590169412023573, −8.823591286066384544964609765432, −8.025600263840239737755999330761, −6.52617123230284920621000946189, −4.96735298161233952164142898068, −3.91285342762432973155287660059, −2.40392633734657348304349419978, −1.14804305772938842096207146812, 1.88075454730649538828532941852, 4.23739862473275538878131488710, 5.24511381819518476455370923540, 6.27548230695885076146624175486, 7.38161315369540023169646876468, 8.272656334490200088131503145574, 8.829886369583236264775668643586, 10.56704401917107635811286803538, 11.27326738733517251479102014527, 12.40125826073179931096854080867

Graph of the ZZ-function along the critical line