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Properties

Label 2-2548-2548.935-c0-0-1
Degree 22
Conductor 25482548
Sign −0.304−0.952i-0.304 - 0.952i
Analytic cond. 1.271611.27161
Root an. cond. 1.127661.12766
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (−0.109 + 1.46i)11-s + (−0.900 + 0.433i)13-s + (0.826 − 0.563i)14-s + (−0.733 + 0.680i)16-s + (−1.63 + 0.246i)17-s + (−0.5 + 0.866i)18-s + (−0.0747 − 0.129i)19-s + (−0.914 + 1.14i)22-s + (0.826 − 0.563i)25-s + (−0.988 − 0.149i)26-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (0.365 + 0.930i)4-s + (0.365 − 0.930i)7-s + (−0.222 + 0.974i)8-s + (0.0747 + 0.997i)9-s + (−0.109 + 1.46i)11-s + (−0.900 + 0.433i)13-s + (0.826 − 0.563i)14-s + (−0.733 + 0.680i)16-s + (−1.63 + 0.246i)17-s + (−0.5 + 0.866i)18-s + (−0.0747 − 0.129i)19-s + (−0.914 + 1.14i)22-s + (0.826 − 0.563i)25-s + (−0.988 − 0.149i)26-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.304−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.304−0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.304−0.952i-0.304 - 0.952i
Analytic conductor: 1.271611.27161
Root analytic conductor: 1.127661.12766
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2548(935,⋅)\chi_{2548} (935, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :0), −0.304−0.952i)(2,\ 2548,\ (\ :0),\ -0.304 - 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8056084381.805608438
L(12)L(\frac12) ≈\approx 1.8056084381.805608438
L(1)L(1) 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+(−0.826−0.563i)T 1 + (-0.826 - 0.563i)T
7 1+(−0.365+0.930i)T 1 + (-0.365 + 0.930i)T
13 1+(0.900−0.433i)T 1 + (0.900 - 0.433i)T
good3 1+(−0.0747−0.997i)T2 1 + (-0.0747 - 0.997i)T^{2}
5 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
11 1+(0.109−1.46i)T+(−0.988−0.149i)T2 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2}
17 1+(1.63−0.246i)T+(0.955−0.294i)T2 1 + (1.63 - 0.246i)T + (0.955 - 0.294i)T^{2}
19 1+(0.0747+0.129i)T+(−0.5+0.866i)T2 1 + (0.0747 + 0.129i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.955−0.294i)T2 1 + (-0.955 - 0.294i)T^{2}
29 1+(−1.19−1.49i)T+(−0.222+0.974i)T2 1 + (-1.19 - 1.49i)T + (-0.222 + 0.974i)T^{2}
31 1+(−0.900+1.56i)T+(−0.5−0.866i)T2 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2}
37 1+(0.733+0.680i)T2 1 + (0.733 + 0.680i)T^{2}
41 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
43 1+(0.900−0.433i)T2 1 + (0.900 - 0.433i)T^{2}
47 1+(−1.57−1.07i)T+(0.365+0.930i)T2 1 + (-1.57 - 1.07i)T + (0.365 + 0.930i)T^{2}
53 1+(0.365+0.930i)T+(−0.733+0.680i)T2 1 + (0.365 + 0.930i)T + (-0.733 + 0.680i)T^{2}
59 1+(−0.698−0.215i)T+(0.826+0.563i)T2 1 + (-0.698 - 0.215i)T + (0.826 + 0.563i)T^{2}
61 1+(0.365−0.930i)T+(−0.733−0.680i)T2 1 + (0.365 - 0.930i)T + (-0.733 - 0.680i)T^{2}
67 1+(−0.988+1.71i)T+(−0.5−0.866i)T2 1 + (-0.988 + 1.71i)T + (-0.5 - 0.866i)T^{2}
71 1+(−1.03+1.29i)T+(−0.222−0.974i)T2 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2}
73 1+(−0.365+0.930i)T2 1 + (-0.365 + 0.930i)T^{2}
79 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
83 1+(1.12+0.541i)T+(0.623+0.781i)T2 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2}
89 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
97 1−T2 1 - 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

−9.175822589085429128347522025918, −8.252967929520591750764922958546, −7.56743630979825643593403302146, −6.97530076446518506643058897335, −6.46474030177280384342597396615, −5.03447121938452896690313881645, −4.60638695759080872027746352034, −4.20367580607025986474439214197, −2.63556677305568919865147736302, −1.98553367067005387598552693473, 0.900459524664371969011678514451, 2.46457111160381990985812761737, 2.93425621707910502329410842077, 4.02870484551053867130170058221, 4.91696918094991722771624761997, 5.62097103889545794188387410877, 6.39109029100605288299766452409, 6.97721115096453849810479962967, 8.403932984561601355059837127270, 8.825308168127772186372812203347

Graph of the ZZ-function along the critical line