↑

Properties

Label 2-3750-5.4-c1-0-47
Degree 22
Conductor 37503750
Sign ii
Analytic cond. 29.943929.9439
Root an. cond. 5.472105.47210
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  − i·2-s − i·3-s − 4-s − 6-s + 0.511i·7-s + i·8-s − 9-s + 3.29·11-s + i·12-s − 2.65i·13-s + 0.511·14-s + 16-s + 5.57i·17-s + i·18-s − 0.374·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.193i·7-s + 0.353i·8-s − 0.333·9-s + 0.993·11-s + 0.288i·12-s − 0.736i·13-s + 0.136·14-s + 0.250·16-s + 1.35i·17-s + 0.235i·18-s − 0.0858·19-s + ⋯

Functional equation

Λ(s)=(3750s/2 ΓC(s) L(s)=(i Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(3750s/2 ΓC(s+1/2) L(s)=(i Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 37503750    =    2⋅3⋅542 \cdot 3 \cdot 5^{4}
Sign: ii
Analytic conductor: 29.943929.9439
Root analytic conductor: 5.472105.47210
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ3750(1249,⋅)\chi_{3750} (1249, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3750, ( :1/2), i)(2,\ 3750,\ (\ :1/2),\ i)

Particular Values

L(1)L(1) ≈\approx 1.8209054171.820905417
L(12)L(\frac12) ≈\approx 1.8209054171.820905417
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)
bad2 1+iT 1 + iT
3 1+iT 1 + iT
5 1 1
good7 1−0.511iT−7T2 1 - 0.511iT - 7T^{2}
11 1−3.29T+11T2 1 - 3.29T + 11T^{2}
13 1+2.65iT−13T2 1 + 2.65iT - 13T^{2}
17 1−5.57iT−17T2 1 - 5.57iT - 17T^{2}
19 1+0.374T+19T2 1 + 0.374T + 19T^{2}
23 1+5.75iT−23T2 1 + 5.75iT - 23T^{2}
29 1−3.89T+29T2 1 - 3.89T + 29T^{2}
31 1+1.80T+31T2 1 + 1.80T + 31T^{2}
37 1−10.3iT−37T2 1 - 10.3iT - 37T^{2}
41 1−2.33T+41T2 1 - 2.33T + 41T^{2}
43 1−5.87iT−43T2 1 - 5.87iT - 43T^{2}
47 1+8.19iT−47T2 1 + 8.19iT - 47T^{2}
53 1+0.518iT−53T2 1 + 0.518iT - 53T^{2}
59 1−7.62T+59T2 1 - 7.62T + 59T^{2}
61 1−2.90T+61T2 1 - 2.90T + 61T^{2}
67 1+1.78iT−67T2 1 + 1.78iT - 67T^{2}
71 1−7.27T+71T2 1 - 7.27T + 71T^{2}
73 1−1.40iT−73T2 1 - 1.40iT - 73T^{2}
79 1−12.1T+79T2 1 - 12.1T + 79T^{2}
83 1+3.09iT−83T2 1 + 3.09iT - 83T^{2}
89 1+3.42T+89T2 1 + 3.42T + 89T^{2}
97 1+15.1iT−97T2 1 + 15.1iT - 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

−8.495379901431532327457101873182, −7.78925011155464039722389356238, −6.67276554530642036611577521353, −6.23186368191093430159511716465, −5.29288665391279195384666230669, −4.34835077029204489018799552978, −3.55157965890240636772148327746, −2.64335299876415013157161082421, −1.70821901860594511549881580449, −0.75415737860635199613155401933, 0.870689289864353836958925763904, 2.30881180570724784172698498669, 3.58655444174073437338873957647, 4.12086573791115325664611324950, 4.99074108151958188207324755906, 5.66975451478769541528546961470, 6.56212099825053014595671521474, 7.17958029264428344252552077613, 7.82752292415562850082480615547, 8.996133401146336027493674689728

Graph of the ZZ-function along the critical line