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Properties

Label 2-1950-13.10-c1-0-19
Degree 22
Conductor 19501950
Sign 0.947+0.319i0.947 + 0.319i
Analytic cond. 15.570815.5708
Root an. cond. 3.945983.94598
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s + (−0.866 − 0.499i)6-s + (2.42 + 1.40i)7-s − 0.999i·8-s + (−0.499 + 0.866i)9-s + (−0.515 + 0.297i)11-s − 0.999·12-s + (3.43 + 1.10i)13-s + 2.80·14-s + (−0.5 − 0.866i)16-s + (−2.87 + 4.98i)17-s + 0.999i·18-s + (6.59 + 3.80i)19-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s + (−0.353 − 0.204i)6-s + (0.918 + 0.530i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.155 + 0.0896i)11-s − 0.288·12-s + (0.951 + 0.307i)13-s + 0.749·14-s + (−0.125 − 0.216i)16-s + (−0.697 + 1.20i)17-s + 0.235i·18-s + (1.51 + 0.873i)19-s + ⋯

Functional equation

Λ(s)=(1950s/2 ΓC(s) L(s)=((0.947+0.319i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1950s/2 ΓC(s+1/2) L(s)=((0.947+0.319i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 19501950    =    2⋅3⋅52⋅132 \cdot 3 \cdot 5^{2} \cdot 13
Sign: 0.947+0.319i0.947 + 0.319i
Analytic conductor: 15.570815.5708
Root analytic conductor: 3.945983.94598
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1950(751,⋅)\chi_{1950} (751, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1950, ( :1/2), 0.947+0.319i)(2,\ 1950,\ (\ :1/2),\ 0.947 + 0.319i)

Particular Values

L(1)L(1) ≈\approx 2.6202077122.620207712
L(12)L(\frac12) ≈\approx 2.6202077122.620207712
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
5 1 1
13 1+(−3.43−1.10i)T 1 + (-3.43 - 1.10i)T
good7 1+(−2.42−1.40i)T+(3.5+6.06i)T2 1 + (-2.42 - 1.40i)T + (3.5 + 6.06i)T^{2}
11 1+(0.515−0.297i)T+(5.5−9.52i)T2 1 + (0.515 - 0.297i)T + (5.5 - 9.52i)T^{2}
17 1+(2.87−4.98i)T+(−8.5−14.7i)T2 1 + (2.87 - 4.98i)T + (-8.5 - 14.7i)T^{2}
19 1+(−6.59−3.80i)T+(9.5+16.4i)T2 1 + (-6.59 - 3.80i)T + (9.5 + 16.4i)T^{2}
23 1+(2.32+4.02i)T+(−11.5+19.9i)T2 1 + (2.32 + 4.02i)T + (-11.5 + 19.9i)T^{2}
29 1+(−1.26−2.18i)T+(−14.5+25.1i)T2 1 + (-1.26 - 2.18i)T + (-14.5 + 25.1i)T^{2}
31 1−6.59iT−31T2 1 - 6.59iT - 31T^{2}
37 1+(−8.98+5.18i)T+(18.5−32.0i)T2 1 + (-8.98 + 5.18i)T + (18.5 - 32.0i)T^{2}
41 1+(4.98−2.87i)T+(20.5−35.5i)T2 1 + (4.98 - 2.87i)T + (20.5 - 35.5i)T^{2}
43 1+(2.12−3.67i)T+(−21.5−37.2i)T2 1 + (2.12 - 3.67i)T + (-21.5 - 37.2i)T^{2}
47 1−2.89iT−47T2 1 - 2.89iT - 47T^{2}
53 1−13.8T+53T2 1 - 13.8T + 53T^{2}
59 1+(8.40+4.85i)T+(29.5+51.0i)T2 1 + (8.40 + 4.85i)T + (29.5 + 51.0i)T^{2}
61 1+(3.41−5.91i)T+(−30.5−52.8i)T2 1 + (3.41 - 5.91i)T + (-30.5 - 52.8i)T^{2}
67 1+(−6.80+3.93i)T+(33.5−58.0i)T2 1 + (-6.80 + 3.93i)T + (33.5 - 58.0i)T^{2}
71 1+(1.11+0.642i)T+(35.5+61.4i)T2 1 + (1.11 + 0.642i)T + (35.5 + 61.4i)T^{2}
73 1+14.5iT−73T2 1 + 14.5iT - 73T^{2}
79 1−1.83T+79T2 1 - 1.83T + 79T^{2}
83 1+4.19iT−83T2 1 + 4.19iT - 83T^{2}
89 1+(−5.24+3.02i)T+(44.5−77.0i)T2 1 + (-5.24 + 3.02i)T + (44.5 - 77.0i)T^{2}
97 1+(14.6+8.45i)T+(48.5+84.0i)T2 1 + (14.6 + 8.45i)T + (48.5 + 84.0i)T^{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

−9.018324510401846746900017196204, −8.314609688435680999512780969392, −7.61772901215829808831823477133, −6.49846122549182997848867245344, −5.93795046011935623499869109800, −5.12001314045449301956005604827, −4.28047664071806896838803657037, −3.25075550653484871823735844093, −2.02657231364230006086304625013, −1.29233410078499573078076585713, 0.932841602075847161944805764354, 2.54706469932172769303645308424, 3.61285302622975875708256716749, 4.43881664585588535630497799334, 5.16209141267444260277024912773, 5.80925237181127970481561734298, 6.86434250549553366705564329465, 7.57555104200419177366494834169, 8.290710354066492136038114746560, 9.240950665223724633677192341318

Graph of the ZZ-function along the critical line