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Properties

Label 2-7e2-7.2-c3-0-5
Degree 22
Conductor 4949
Sign 0.701+0.712i0.701 + 0.712i
Analytic cond. 2.891092.89109
Root an. cond. 1.700321.70032
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  + (0.5 − 0.866i)2-s + (−1 − 1.73i)3-s + (3.5 + 6.06i)4-s + (8 − 13.8i)5-s − 1.99·6-s + 15·8-s + (11.5 − 19.9i)9-s + (−7.99 − 13.8i)10-s + (4 + 6.92i)11-s + (7 − 12.1i)12-s − 28·13-s − 31.9·15-s + (−20.5 + 35.5i)16-s + (27 + 46.7i)17-s + (−11.5 − 19.9i)18-s + (−55 + 95.2i)19-s + ⋯
L(s)  = 1  + (0.176 − 0.306i)2-s + (−0.192 − 0.333i)3-s + (0.437 + 0.757i)4-s + (0.715 − 1.23i)5-s − 0.136·6-s + 0.662·8-s + (0.425 − 0.737i)9-s + (−0.252 − 0.438i)10-s + (0.109 + 0.189i)11-s + (0.168 − 0.291i)12-s − 0.597·13-s − 0.550·15-s + (−0.320 + 0.554i)16-s + (0.385 + 0.667i)17-s + (−0.150 − 0.260i)18-s + (−0.664 + 1.15i)19-s + ⋯

Functional equation

Λ(s)=(49s/2 ΓC(s) L(s)=((0.701+0.712i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(49s/2 ΓC(s+3/2) L(s)=((0.701+0.712i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 4949    =    727^{2}
Sign: 0.701+0.712i0.701 + 0.712i
Analytic conductor: 2.891092.89109
Root analytic conductor: 1.700321.70032
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ49(30,⋅)\chi_{49} (30, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 49, ( :3/2), 0.701+0.712i)(2,\ 49,\ (\ :3/2),\ 0.701 + 0.712i)

Particular Values

L(2)L(2) ≈\approx 1.54483−0.647413i1.54483 - 0.647413i
L(12)L(\frac12) ≈\approx 1.54483−0.647413i1.54483 - 0.647413i
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)
bad7 1 1
good2 1+(−0.5+0.866i)T+(−4−6.92i)T2 1 + (-0.5 + 0.866i)T + (-4 - 6.92i)T^{2}
3 1+(1+1.73i)T+(−13.5+23.3i)T2 1 + (1 + 1.73i)T + (-13.5 + 23.3i)T^{2}
5 1+(−8+13.8i)T+(−62.5−108.i)T2 1 + (-8 + 13.8i)T + (-62.5 - 108. i)T^{2}
11 1+(−4−6.92i)T+(−665.5+1.15e3i)T2 1 + (-4 - 6.92i)T + (-665.5 + 1.15e3i)T^{2}
13 1+28T+2.19e3T2 1 + 28T + 2.19e3T^{2}
17 1+(−27−46.7i)T+(−2.45e3+4.25e3i)T2 1 + (-27 - 46.7i)T + (-2.45e3 + 4.25e3i)T^{2}
19 1+(55−95.2i)T+(−3.42e3−5.94e3i)T2 1 + (55 - 95.2i)T + (-3.42e3 - 5.94e3i)T^{2}
23 1+(24−41.5i)T+(−6.08e3−1.05e4i)T2 1 + (24 - 41.5i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+110T+2.43e4T2 1 + 110T + 2.43e4T^{2}
31 1+(−6−10.3i)T+(−1.48e4+2.57e4i)T2 1 + (-6 - 10.3i)T + (-1.48e4 + 2.57e4i)T^{2}
37 1+(−123+213.i)T+(−2.53e4−4.38e4i)T2 1 + (-123 + 213. i)T + (-2.53e4 - 4.38e4i)T^{2}
41 1+182T+6.89e4T2 1 + 182T + 6.89e4T^{2}
43 1−128T+7.95e4T2 1 - 128T + 7.95e4T^{2}
47 1+(−162+280.i)T+(−5.19e4−8.99e4i)T2 1 + (-162 + 280. i)T + (-5.19e4 - 8.99e4i)T^{2}
53 1+(−81−140.i)T+(−7.44e4+1.28e5i)T2 1 + (-81 - 140. i)T + (-7.44e4 + 1.28e5i)T^{2}
59 1+(−405−701.i)T+(−1.02e5+1.77e5i)T2 1 + (-405 - 701. i)T + (-1.02e5 + 1.77e5i)T^{2}
61 1+(244−422.i)T+(−1.13e5−1.96e5i)T2 1 + (244 - 422. i)T + (-1.13e5 - 1.96e5i)T^{2}
67 1+(122+211.i)T+(−1.50e5+2.60e5i)T2 1 + (122 + 211. i)T + (-1.50e5 + 2.60e5i)T^{2}
71 1+768T+3.57e5T2 1 + 768T + 3.57e5T^{2}
73 1+(351+607.i)T+(−1.94e5+3.36e5i)T2 1 + (351 + 607. i)T + (-1.94e5 + 3.36e5i)T^{2}
79 1+(220−381.i)T+(−2.46e5−4.26e5i)T2 1 + (220 - 381. i)T + (-2.46e5 - 4.26e5i)T^{2}
83 1−1.30e3T+5.71e5T2 1 - 1.30e3T + 5.71e5T^{2}
89 1+(−365+632.i)T+(−3.52e5−6.10e5i)T2 1 + (-365 + 632. i)T + (-3.52e5 - 6.10e5i)T^{2}
97 1+294T+9.12e5T2 1 + 294T + 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

−14.95928764465242081501564011250, −13.34551402561014419930281042466, −12.54759605560259208078556604664, −11.98062440207555109203400991660, −10.20260110566653581829369896777, −8.883811470031736088011551101749, −7.48518103583452484772492699557, −5.88548694663386714316467601804, −4.07313329170699156517493870062, −1.70256716586223987954185963424, 2.38851112186511548991445905713, 4.96186071834753755412435917722, 6.33374777178742990390553892608, 7.36064990897455679231355987419, 9.678773948849315475126704849014, 10.51231436876016165329947925052, 11.34566488586736534784652419080, 13.36103545347305961138218382611, 14.32159472098490108690740481478, 15.12449547211677895848558907652

Graph of the ZZ-function along the critical line