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Properties

Label 2-1805-5.4-c1-0-19
Degree 22
Conductor 18051805
Sign 0.914+0.403i0.914 + 0.403i
Analytic cond. 14.412914.4129
Root an. cond. 3.796443.79644
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  + 1.96i·2-s + 3.10i·3-s − 1.87·4-s + (−2.04 − 0.902i)5-s − 6.10·6-s + 2.84i·7-s + 0.255i·8-s − 6.62·9-s + (1.77 − 4.02i)10-s − 0.295·11-s − 5.80i·12-s + 2.61i·13-s − 5.59·14-s + (2.80 − 6.34i)15-s − 4.24·16-s + 7.09i·17-s + ⋯
L(s)  = 1  + 1.39i·2-s + 1.79i·3-s − 0.935·4-s + (−0.914 − 0.403i)5-s − 2.49·6-s + 1.07i·7-s + 0.0904i·8-s − 2.20·9-s + (0.561 − 1.27i)10-s − 0.0891·11-s − 1.67i·12-s + 0.726i·13-s − 1.49·14-s + (0.723 − 1.63i)15-s − 1.06·16-s + 1.72i·17-s + ⋯

Functional equation

Λ(s)=(1805s/2 ΓC(s) L(s)=((0.914+0.403i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1805s/2 ΓC(s+1/2) L(s)=((0.914+0.403i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18051805    =    5⋅1925 \cdot 19^{2}
Sign: 0.914+0.403i0.914 + 0.403i
Analytic conductor: 14.412914.4129
Root analytic conductor: 3.796443.79644
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1805(1084,⋅)\chi_{1805} (1084, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1805, ( :1/2), 0.914+0.403i)(2,\ 1805,\ (\ :1/2),\ 0.914 + 0.403i)

Particular Values

L(1)L(1) ≈\approx 0.96612893690.9661289369
L(12)L(\frac12) ≈\approx 0.96612893690.9661289369
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+(2.04+0.902i)T 1 + (2.04 + 0.902i)T
19 1 1
good2 1−1.96iT−2T2 1 - 1.96iT - 2T^{2}
3 1−3.10iT−3T2 1 - 3.10iT - 3T^{2}
7 1−2.84iT−7T2 1 - 2.84iT - 7T^{2}
11 1+0.295T+11T2 1 + 0.295T + 11T^{2}
13 1−2.61iT−13T2 1 - 2.61iT - 13T^{2}
17 1−7.09iT−17T2 1 - 7.09iT - 17T^{2}
23 1+2.66iT−23T2 1 + 2.66iT - 23T^{2}
29 1−1.25T+29T2 1 - 1.25T + 29T^{2}
31 1+1.74T+31T2 1 + 1.74T + 31T^{2}
37 1+0.722iT−37T2 1 + 0.722iT - 37T^{2}
41 1−10.1T+41T2 1 - 10.1T + 41T^{2}
43 1+4.02iT−43T2 1 + 4.02iT - 43T^{2}
47 1−2.94iT−47T2 1 - 2.94iT - 47T^{2}
53 1−6.98iT−53T2 1 - 6.98iT - 53T^{2}
59 1−8.84T+59T2 1 - 8.84T + 59T^{2}
61 1−6.62T+61T2 1 - 6.62T + 61T^{2}
67 1−1.93iT−67T2 1 - 1.93iT - 67T^{2}
71 1−15.7T+71T2 1 - 15.7T + 71T^{2}
73 1−3.05iT−73T2 1 - 3.05iT - 73T^{2}
79 1+8.06T+79T2 1 + 8.06T + 79T^{2}
83 1−13.8iT−83T2 1 - 13.8iT - 83T^{2}
89 1+0.551T+89T2 1 + 0.551T + 89T^{2}
97 1+6.60iT−97T2 1 + 6.60iT - 97T^{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

−9.647401768018907330618434178279, −8.896261050641072507621682543463, −8.557553247990617935050889432676, −7.905897898779127673809379712383, −6.69076205761881642647903345486, −5.77048466681879299093384555108, −5.28947222549819724501358760361, −4.32627264280267077715005768145, −3.90726235914782008766432699283, −2.51547647605495985868180261972, 0.43992828745667645433170694850, 1.00113713789487724328389053992, 2.35391147889967309754566936944, 3.04913661453041155435226341696, 3.89795409380461363214272818903, 5.12309365536705136151505086283, 6.48139726091572628636053961492, 7.21170587320951458058187994797, 7.54014427809725378113654945715, 8.382144135187216178105541033290

Graph of the ZZ-function along the critical line