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Properties

Label 2-75-5.4-c3-0-1
Degree 22
Conductor 7575
Sign −0.894−0.447i-0.894 - 0.447i
Analytic cond. 4.425144.42514
Root an. cond. 2.103602.10360
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s + 3i·3-s − 4-s − 9·6-s + 20i·7-s + 21i·8-s − 9·9-s − 24·11-s − 3i·12-s − 74i·13-s − 60·14-s − 71·16-s + 54i·17-s − 27i·18-s + 124·19-s + ⋯
L(s)  = 1  + 1.06i·2-s + 0.577i·3-s − 0.125·4-s − 0.612·6-s + 1.07i·7-s + 0.928i·8-s − 0.333·9-s − 0.657·11-s − 0.0721i·12-s − 1.57i·13-s − 1.14·14-s − 1.10·16-s + 0.770i·17-s − 0.353i·18-s + 1.49·19-s + ⋯

Functional equation

Λ(s)=(75s/2 ΓC(s) L(s)=((−0.894−0.447i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(75s/2 ΓC(s+3/2) L(s)=((−0.894−0.447i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7575    =    3⋅523 \cdot 5^{2}
Sign: −0.894−0.447i-0.894 - 0.447i
Analytic conductor: 4.425144.42514
Root analytic conductor: 2.103602.10360
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ75(49,⋅)\chi_{75} (49, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 75, ( :3/2), −0.894−0.447i)(2,\ 75,\ (\ :3/2),\ -0.894 - 0.447i)

Particular Values

L(2)L(2) ≈\approx 0.340841+1.44382i0.340841 + 1.44382i
L(12)L(\frac12) ≈\approx 0.340841+1.44382i0.340841 + 1.44382i
L(52)L(\frac{5}{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 1−3iT 1 - 3iT
5 1 1
good2 1−3iT−8T2 1 - 3iT - 8T^{2}
7 1−20iT−343T2 1 - 20iT - 343T^{2}
11 1+24T+1.33e3T2 1 + 24T + 1.33e3T^{2}
13 1+74iT−2.19e3T2 1 + 74iT - 2.19e3T^{2}
17 1−54iT−4.91e3T2 1 - 54iT - 4.91e3T^{2}
19 1−124T+6.85e3T2 1 - 124T + 6.85e3T^{2}
23 1−120iT−1.21e4T2 1 - 120iT - 1.21e4T^{2}
29 1−78T+2.43e4T2 1 - 78T + 2.43e4T^{2}
31 1−200T+2.97e4T2 1 - 200T + 2.97e4T^{2}
37 1+70iT−5.06e4T2 1 + 70iT - 5.06e4T^{2}
41 1−330T+6.89e4T2 1 - 330T + 6.89e4T^{2}
43 1+92iT−7.95e4T2 1 + 92iT - 7.95e4T^{2}
47 1+24iT−1.03e5T2 1 + 24iT - 1.03e5T^{2}
53 1+450iT−1.48e5T2 1 + 450iT - 1.48e5T^{2}
59 1+24T+2.05e5T2 1 + 24T + 2.05e5T^{2}
61 1+322T+2.26e5T2 1 + 322T + 2.26e5T^{2}
67 1+196iT−3.00e5T2 1 + 196iT - 3.00e5T^{2}
71 1+288T+3.57e5T2 1 + 288T + 3.57e5T^{2}
73 1−430iT−3.89e5T2 1 - 430iT - 3.89e5T^{2}
79 1−520T+4.93e5T2 1 - 520T + 4.93e5T^{2}
83 1+156iT−5.71e5T2 1 + 156iT - 5.71e5T^{2}
89 1+1.02e3T+7.04e5T2 1 + 1.02e3T + 7.04e5T^{2}
97 1+286iT−9.12e5T2 1 + 286iT - 9.12e5T^{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

−15.03125201256977245588962213195, −13.73945717343954403496584134429, −12.34464803859620993708393201461, −11.15298426659519750706286487841, −9.889490218891267285911835915667, −8.499823731422232753226216326681, −7.63597275054706864707941696511, −5.86980239748208835300091317570, −5.25187529220978039219541695512, −2.89293842875589818093107360515, 1.01638563299130841948113444224, 2.73607558073614867963940151912, 4.42027967002133456287907778075, 6.62312989728905323968477608204, 7.56623647510769697029388780433, 9.357609616257843249023703694968, 10.45087068764999500448914841418, 11.47081566805891243564620641004, 12.25758712245231039242659271315, 13.52468146302563541010408213241

Graph of the ZZ-function along the critical line