↑

Properties

Label 2-755-5.4-c1-0-53
Degree 22
Conductor 755755
Sign 0.543+0.839i0.543 + 0.839i
Analytic cond. 6.028706.02870
Root an. cond. 2.455342.45534
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.783i·2-s + 2.94i·3-s + 1.38·4-s + (−1.21 − 1.87i)5-s + 2.30·6-s − 2.34i·7-s − 2.65i·8-s − 5.66·9-s + (−1.47 + 0.952i)10-s + 3.68·11-s + 4.08i·12-s − 5.30i·13-s − 1.83·14-s + (5.52 − 3.57i)15-s + 0.694·16-s − 1.46i·17-s + ⋯
L(s)  = 1  − 0.553i·2-s + 1.69i·3-s + 0.693·4-s + (−0.543 − 0.839i)5-s + 0.941·6-s − 0.887i·7-s − 0.937i·8-s − 1.88·9-s + (−0.464 + 0.301i)10-s + 1.10·11-s + 1.17i·12-s − 1.47i·13-s − 0.491·14-s + (1.42 − 0.923i)15-s + 0.173·16-s − 0.354i·17-s + ⋯

Functional equation

Λ(s)=(755s/2 ΓC(s) L(s)=((0.543+0.839i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(755s/2 ΓC(s+1/2) L(s)=((0.543+0.839i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 + 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 755755    =    5⋅1515 \cdot 151
Sign: 0.543+0.839i0.543 + 0.839i
Analytic conductor: 6.028706.02870
Root analytic conductor: 2.455342.45534
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ755(454,⋅)\chi_{755} (454, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 755, ( :1/2), 0.543+0.839i)(2,\ 755,\ (\ :1/2),\ 0.543 + 0.839i)

Particular Values

L(1)L(1) ≈\approx 1.37197−0.746032i1.37197 - 0.746032i
L(12)L(\frac12) ≈\approx 1.37197−0.746032i1.37197 - 0.746032i
L(32)L(\frac{3}{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)
bad5 1+(1.21+1.87i)T 1 + (1.21 + 1.87i)T
151 1+T 1 + T
good2 1+0.783iT−2T2 1 + 0.783iT - 2T^{2}
3 1−2.94iT−3T2 1 - 2.94iT - 3T^{2}
7 1+2.34iT−7T2 1 + 2.34iT - 7T^{2}
11 1−3.68T+11T2 1 - 3.68T + 11T^{2}
13 1+5.30iT−13T2 1 + 5.30iT - 13T^{2}
17 1+1.46iT−17T2 1 + 1.46iT - 17T^{2}
19 1+4.99T+19T2 1 + 4.99T + 19T^{2}
23 1+6.83iT−23T2 1 + 6.83iT - 23T^{2}
29 1+8.58T+29T2 1 + 8.58T + 29T^{2}
31 1−6.59T+31T2 1 - 6.59T + 31T^{2}
37 1−6.57iT−37T2 1 - 6.57iT - 37T^{2}
41 1−10.8T+41T2 1 - 10.8T + 41T^{2}
43 1−1.24iT−43T2 1 - 1.24iT - 43T^{2}
47 1−3.49iT−47T2 1 - 3.49iT - 47T^{2}
53 1+1.36iT−53T2 1 + 1.36iT - 53T^{2}
59 1−5.00T+59T2 1 - 5.00T + 59T^{2}
61 1−11.3T+61T2 1 - 11.3T + 61T^{2}
67 1−1.06iT−67T2 1 - 1.06iT - 67T^{2}
71 1+5.02T+71T2 1 + 5.02T + 71T^{2}
73 1−15.9iT−73T2 1 - 15.9iT - 73T^{2}
79 1−2.84T+79T2 1 - 2.84T + 79T^{2}
83 1+2.19iT−83T2 1 + 2.19iT - 83T^{2}
89 1+5.94T+89T2 1 + 5.94T + 89T^{2}
97 1−16.2iT−97T2 1 - 16.2iT - 97T^{2}
show more
show less
   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

−10.28341362507832426223048080772, −9.661266275949201464338265215943, −8.718765291466413101139495973665, −7.86707193853078229156693166798, −6.61570542021040226332160026543, −5.48012505769513636013688151389, −4.24041381509944477976540223348, −3.98453044381737651872297960028, −2.82653778962239593651051807104, −0.796312388738635197760515432436, 1.78852055122684605026590668104, 2.40163058280959169318923681906, 3.87494555584778473951137071487, 5.82584237223109722144280919434, 6.31256190257174540546429625854, 7.01275716739345879734088449858, 7.53204097063189390666933151153, 8.459565058675370577065195910040, 9.260009585466749715550536760910, 10.93428720178776456296950457538

Graph of the ZZ-function along the critical line