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Properties

Label 2-1014-13.9-c1-0-2
Degree 22
Conductor 10141014
Sign −0.999−0.0256i-0.999 - 0.0256i
Analytic cond. 8.096838.09683
Root an. cond. 2.845492.84549
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s − 2·5-s + (0.499 + 0.866i)6-s + (2 + 3.46i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)10-s + (−2 + 3.46i)11-s − 0.999·12-s − 3.99·14-s + (−1 + 1.73i)15-s + (−0.5 + 0.866i)16-s + (−1 − 1.73i)17-s + 0.999·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s − 0.894·5-s + (0.204 + 0.353i)6-s + (0.755 + 1.30i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.316 − 0.547i)10-s + (−0.603 + 1.04i)11-s − 0.288·12-s − 1.06·14-s + (−0.258 + 0.447i)15-s + (−0.125 + 0.216i)16-s + (−0.242 − 0.420i)17-s + 0.235·18-s + ⋯

Functional equation

Λ(s)=(1014s/2 ΓC(s) L(s)=((−0.999−0.0256i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1014s/2 ΓC(s+1/2) L(s)=((−0.999−0.0256i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10141014    =    2⋅3⋅1322 \cdot 3 \cdot 13^{2}
Sign: −0.999−0.0256i-0.999 - 0.0256i
Analytic conductor: 8.096838.09683
Root analytic conductor: 2.845492.84549
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1014(529,⋅)\chi_{1014} (529, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1014, ( :1/2), −0.999−0.0256i)(2,\ 1014,\ (\ :1/2),\ -0.999 - 0.0256i)

Particular Values

L(1)L(1) ≈\approx 0.40227854640.4022785464
L(12)L(\frac12) ≈\approx 0.40227854640.4022785464
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
13 1 1
good5 1+2T+5T2 1 + 2T + 5T^{2}
7 1+(−2−3.46i)T+(−3.5+6.06i)T2 1 + (-2 - 3.46i)T + (-3.5 + 6.06i)T^{2}
11 1+(2−3.46i)T+(−5.5−9.52i)T2 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2}
17 1+(1+1.73i)T+(−8.5+14.7i)T2 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2}
19 1+(4+6.92i)T+(−9.5+16.4i)T2 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2}
23 1+(−11.5−19.9i)T2 1 + (-11.5 - 19.9i)T^{2}
29 1+(3−5.19i)T+(−14.5−25.1i)T2 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2}
31 1−4T+31T2 1 - 4T + 31T^{2}
37 1+(1−1.73i)T+(−18.5−32.0i)T2 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2}
41 1+(5−8.66i)T+(−20.5−35.5i)T2 1 + (5 - 8.66i)T + (-20.5 - 35.5i)T^{2}
43 1+(2+3.46i)T+(−21.5+37.2i)T2 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+10T+53T2 1 + 10T + 53T^{2}
59 1+(−2−3.46i)T+(−29.5+51.0i)T2 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2}
61 1+(−1−1.73i)T+(−30.5+52.8i)T2 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2}
67 1+(8−13.8i)T+(−33.5−58.0i)T2 1 + (8 - 13.8i)T + (-33.5 - 58.0i)T^{2}
71 1+(4+6.92i)T+(−35.5+61.4i)T2 1 + (4 + 6.92i)T + (-35.5 + 61.4i)T^{2}
73 1+2T+73T2 1 + 2T + 73T^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+(−7+12.1i)T+(−44.5−77.0i)T2 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2}
97 1+(−5−8.66i)T+(−48.5+84.0i)T2 1 + (-5 - 8.66i)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

−10.23293368755067370434529154658, −9.145368391691606469633274913217, −8.595900598507780154457311413929, −7.894324956602114837988826055505, −7.18880109604546421318558008814, −6.35877488863836362857121327480, −5.09106181565756656540423357970, −4.58148990842437095798915005268, −2.87820913814768552798286213066, −1.83989499757516797369505272854, 0.19250200304708857246001916242, 1.80490141972538580177281045757, 3.41033223829196181202537127968, 3.94690539412875462211810166483, 4.75662507632535100368520916332, 6.10774924336476300793289181373, 7.49152982329765896674352078523, 8.135884460639258980653108170615, 8.437538714039520507205622594830, 9.798364260814972238884862418113

Graph of the ZZ-function along the critical line