Properties

Label 2-755-5.4-c1-0-62
Degree 22
Conductor 755755
Sign 0.105+0.994i0.105 + 0.994i
Analytic cond. 6.028706.02870
Root an. cond. 2.455342.45534
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  − 0.250i·2-s + 0.442i·3-s + 1.93·4-s + (−0.235 − 2.22i)5-s + 0.110·6-s − 1.63i·7-s − 0.985i·8-s + 2.80·9-s + (−0.556 + 0.0588i)10-s − 5.60·11-s + 0.858i·12-s − 1.36i·13-s − 0.410·14-s + (0.984 − 0.104i)15-s + 3.62·16-s − 6.74i·17-s + ⋯
L(s)  = 1  − 0.176i·2-s + 0.255i·3-s + 0.968·4-s + (−0.105 − 0.994i)5-s + 0.0452·6-s − 0.619i·7-s − 0.348i·8-s + 0.934·9-s + (−0.175 + 0.0186i)10-s − 1.69·11-s + 0.247i·12-s − 0.377i·13-s − 0.109·14-s + (0.254 − 0.0269i)15-s + 0.907·16-s − 1.63i·17-s + ⋯

Functional equation

Λ(s)=(755s/2ΓC(s)L(s)=((0.105+0.994i)Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(755s/2ΓC(s+1/2)L(s)=((0.105+0.994i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 755755    =    51515 \cdot 151
Sign: 0.105+0.994i0.105 + 0.994i
Analytic conductor: 6.028706.02870
Root analytic conductor: 2.455342.45534
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ755(454,)\chi_{755} (454, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 755, ( :1/2), 0.105+0.994i)(2,\ 755,\ (\ :1/2),\ 0.105 + 0.994i)

Particular Values

L(1)L(1) \approx 1.303731.17307i1.30373 - 1.17307i
L(12)L(\frac12) \approx 1.303731.17307i1.30373 - 1.17307i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1+(0.235+2.22i)T 1 + (0.235 + 2.22i)T
151 1+T 1 + T
good2 1+0.250iT2T2 1 + 0.250iT - 2T^{2}
3 10.442iT3T2 1 - 0.442iT - 3T^{2}
7 1+1.63iT7T2 1 + 1.63iT - 7T^{2}
11 1+5.60T+11T2 1 + 5.60T + 11T^{2}
13 1+1.36iT13T2 1 + 1.36iT - 13T^{2}
17 1+6.74iT17T2 1 + 6.74iT - 17T^{2}
19 1+0.591T+19T2 1 + 0.591T + 19T^{2}
23 12.58iT23T2 1 - 2.58iT - 23T^{2}
29 16.70T+29T2 1 - 6.70T + 29T^{2}
31 1+7.88T+31T2 1 + 7.88T + 31T^{2}
37 13.52iT37T2 1 - 3.52iT - 37T^{2}
41 13.19T+41T2 1 - 3.19T + 41T^{2}
43 111.8iT43T2 1 - 11.8iT - 43T^{2}
47 1+7.93iT47T2 1 + 7.93iT - 47T^{2}
53 1+11.3iT53T2 1 + 11.3iT - 53T^{2}
59 14.11T+59T2 1 - 4.11T + 59T^{2}
61 18.31T+61T2 1 - 8.31T + 61T^{2}
67 1+9.36iT67T2 1 + 9.36iT - 67T^{2}
71 111.6T+71T2 1 - 11.6T + 71T^{2}
73 111.1iT73T2 1 - 11.1iT - 73T^{2}
79 15.19T+79T2 1 - 5.19T + 79T^{2}
83 13.94iT83T2 1 - 3.94iT - 83T^{2}
89 1+17.8T+89T2 1 + 17.8T + 89T^{2}
97 1+5.09iT97T2 1 + 5.09iT - 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−10.08981509283661374327200975661, −9.641738322849345904766837406760, −8.238522800705814154473124737035, −7.55478654190111491512224844364, −6.87230556949962694614744806653, −5.41747964002810975107737488506, −4.82505010637232825577023002590, −3.55754449868791099597288543473, −2.34643231180975377318999023376, −0.877519970642938233558487678818, 1.98121494225564122066304245614, 2.67607339406946285849369037504, 3.98093292909405736546090966146, 5.50452271503474316034319434819, 6.25019920753567665547816924157, 7.12197151674008213981217392125, 7.72019300868299310392055525197, 8.582676636538905531110823419234, 10.07939603682858956974717668972, 10.59354254344767668451852103257

Graph of the ZZ-function along the critical line