Properties

Label 2-3332-3332.2515-c0-0-0
Degree 22
Conductor 33323332
Sign 0.481+0.876i-0.481 + 0.876i
Analytic cond. 1.662881.66288
Root an. cond. 1.289521.28952
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.149 − 1.98i)3-s + (−0.733 + 0.680i)4-s + (−1.79 + 0.865i)6-s + (−0.433 + 0.900i)7-s + (0.900 + 0.433i)8-s + (−2.94 + 0.443i)9-s + (1.84 + 0.277i)11-s + (1.46 + 1.35i)12-s + (−0.455 + 0.571i)13-s + (0.997 + 0.0747i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (1.48 + 2.57i)18-s + (1.85 + 0.728i)21-s + (−0.414 − 1.81i)22-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)2-s + (−0.149 − 1.98i)3-s + (−0.733 + 0.680i)4-s + (−1.79 + 0.865i)6-s + (−0.433 + 0.900i)7-s + (0.900 + 0.433i)8-s + (−2.94 + 0.443i)9-s + (1.84 + 0.277i)11-s + (1.46 + 1.35i)12-s + (−0.455 + 0.571i)13-s + (0.997 + 0.0747i)14-s + (0.0747 − 0.997i)16-s + (0.955 − 0.294i)17-s + (1.48 + 2.57i)18-s + (1.85 + 0.728i)21-s + (−0.414 − 1.81i)22-s + ⋯

Functional equation

Λ(s)=(3332s/2ΓC(s)L(s)=((0.481+0.876i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3332s/2ΓC(s)L(s)=((0.481+0.876i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 33323332    =    2272172^{2} \cdot 7^{2} \cdot 17
Sign: 0.481+0.876i-0.481 + 0.876i
Analytic conductor: 1.662881.66288
Root analytic conductor: 1.289521.28952
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3332(2515,)\chi_{3332} (2515, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3332, ( :0), 0.481+0.876i)(2,\ 3332,\ (\ :0),\ -0.481 + 0.876i)

Particular Values

L(12)L(\frac{1}{2}) \approx 0.92498349980.9249834998
L(12)L(\frac12) \approx 0.92498349980.9249834998
L(1)L(1) 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)
bad2 1+(0.365+0.930i)T 1 + (0.365 + 0.930i)T
7 1+(0.4330.900i)T 1 + (0.433 - 0.900i)T
17 1+(0.955+0.294i)T 1 + (-0.955 + 0.294i)T
good3 1+(0.149+1.98i)T+(0.988+0.149i)T2 1 + (0.149 + 1.98i)T + (-0.988 + 0.149i)T^{2}
5 1+(0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
11 1+(1.840.277i)T+(0.955+0.294i)T2 1 + (-1.84 - 0.277i)T + (0.955 + 0.294i)T^{2}
13 1+(0.4550.571i)T+(0.2220.974i)T2 1 + (0.455 - 0.571i)T + (-0.222 - 0.974i)T^{2}
19 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
23 1+(1.490.460i)T+(0.826+0.563i)T2 1 + (-1.49 - 0.460i)T + (0.826 + 0.563i)T^{2}
29 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
31 1+(0.4330.751i)T+(0.5+0.866i)T2 1 + (-0.433 - 0.751i)T + (-0.5 + 0.866i)T^{2}
37 1+(0.07470.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
41 1+(0.6230.781i)T2 1 + (-0.623 - 0.781i)T^{2}
43 1+(0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
47 1+(0.7330.680i)T2 1 + (0.733 - 0.680i)T^{2}
53 1+(1.211.12i)T+(0.07470.997i)T2 1 + (1.21 - 1.12i)T + (0.0747 - 0.997i)T^{2}
59 1+(0.3650.930i)T2 1 + (-0.365 - 0.930i)T^{2}
61 1+(0.07470.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
67 1+(0.50.866i)T2 1 + (0.5 - 0.866i)T^{2}
71 1+(0.2501.09i)T+(0.900+0.433i)T2 1 + (-0.250 - 1.09i)T + (-0.900 + 0.433i)T^{2}
73 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
79 1+(0.149+0.258i)T+(0.50.866i)T2 1 + (-0.149 + 0.258i)T + (-0.5 - 0.866i)T^{2}
83 1+(0.2220.974i)T2 1 + (0.222 - 0.974i)T^{2}
89 1+(0.147+0.0222i)T+(0.9550.294i)T2 1 + (-0.147 + 0.0222i)T + (0.955 - 0.294i)T^{2}
97 1T2 1 - T^{2}
show more
show less
   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

−8.754544724297870351345490955772, −7.84738728630060470723852843624, −7.02651647684643329954788268327, −6.60972766400120352272251827049, −5.70803714724611779299667061271, −4.74191036026273193172260542667, −3.31606604418325077881974066859, −2.69026325392106622771774940414, −1.71231673097765192992542225312, −1.03774895530056787745488036532, 0.884543582747539532695854431845, 3.27379726588034037706103030928, 3.72050301161578072520764466380, 4.57284103610135599657559904490, 5.14783706370095813021331311642, 6.05739409530682606607523932561, 6.63020090545962215425442197434, 7.63924988433357425389075627848, 8.556485924309393959996671871375, 9.191474060021538723185689281640

Graph of the ZZ-function along the critical line