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Properties

Label 2-1323-49.45-c0-0-0
Degree 22
Conductor 13231323
Sign 0.304+0.952i0.304 + 0.952i
Analytic cond. 0.6602630.660263
Root an. cond. 0.8125650.812565
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  + (−0.365 − 0.930i)4-s + (0.900 − 0.433i)7-s + (−0.255 − 0.531i)13-s + (−0.733 + 0.680i)16-s + (0.751 − 0.433i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.0332 + 0.145i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 0.432i)52-s + (0.277 + 0.108i)61-s + (0.900 + 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)4-s + (0.900 − 0.433i)7-s + (−0.255 − 0.531i)13-s + (−0.733 + 0.680i)16-s + (0.751 − 0.433i)19-s + (0.826 − 0.563i)25-s + (−0.733 − 0.680i)28-s + (−1.61 − 0.930i)31-s + (−0.266 + 0.680i)37-s + (0.0332 + 0.145i)43-s + (0.623 − 0.781i)49-s + (−0.400 + 0.432i)52-s + (0.277 + 0.108i)61-s + (0.900 + 0.433i)64-s + (0.733 − 1.26i)67-s + ⋯

Functional equation

Λ(s)=(1323s/2 ΓC(s) L(s)=((0.304+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1323s/2 ΓC(s) L(s)=((0.304+0.952i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 13231323    =    33⋅723^{3} \cdot 7^{2}
Sign: 0.304+0.952i0.304 + 0.952i
Analytic conductor: 0.6602630.660263
Root analytic conductor: 0.8125650.812565
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1323(1270,⋅)\chi_{1323} (1270, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1323, ( :0), 0.304+0.952i)(2,\ 1323,\ (\ :0),\ 0.304 + 0.952i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.0469688241.046968824
L(12)L(\frac12) ≈\approx 1.0469688241.046968824
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)
bad3 1 1
7 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
good2 1+(0.365+0.930i)T2 1 + (0.365 + 0.930i)T^{2}
5 1+(−0.826+0.563i)T2 1 + (-0.826 + 0.563i)T^{2}
11 1+(−0.988−0.149i)T2 1 + (-0.988 - 0.149i)T^{2}
13 1+(0.255+0.531i)T+(−0.623+0.781i)T2 1 + (0.255 + 0.531i)T + (-0.623 + 0.781i)T^{2}
17 1+(−0.955+0.294i)T2 1 + (-0.955 + 0.294i)T^{2}
19 1+(−0.751+0.433i)T+(0.5−0.866i)T2 1 + (-0.751 + 0.433i)T + (0.5 - 0.866i)T^{2}
23 1+(0.955+0.294i)T2 1 + (0.955 + 0.294i)T^{2}
29 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
31 1+(1.61+0.930i)T+(0.5+0.866i)T2 1 + (1.61 + 0.930i)T + (0.5 + 0.866i)T^{2}
37 1+(0.266−0.680i)T+(−0.733−0.680i)T2 1 + (0.266 - 0.680i)T + (-0.733 - 0.680i)T^{2}
41 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
43 1+(−0.0332−0.145i)T+(−0.900+0.433i)T2 1 + (-0.0332 - 0.145i)T + (-0.900 + 0.433i)T^{2}
47 1+(−0.365−0.930i)T2 1 + (-0.365 - 0.930i)T^{2}
53 1+(−0.733+0.680i)T2 1 + (-0.733 + 0.680i)T^{2}
59 1+(−0.826−0.563i)T2 1 + (-0.826 - 0.563i)T^{2}
61 1+(−0.277−0.108i)T+(0.733+0.680i)T2 1 + (-0.277 - 0.108i)T + (0.733 + 0.680i)T^{2}
67 1+(−0.733+1.26i)T+(−0.5−0.866i)T2 1 + (-0.733 + 1.26i)T + (-0.5 - 0.866i)T^{2}
71 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
73 1+(0.880+1.29i)T+(−0.365+0.930i)T2 1 + (0.880 + 1.29i)T + (-0.365 + 0.930i)T^{2}
79 1+(−0.988−1.71i)T+(−0.5+0.866i)T2 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2}
83 1+(−0.623−0.781i)T2 1 + (-0.623 - 0.781i)T^{2}
89 1+(0.988−0.149i)T2 1 + (0.988 - 0.149i)T^{2}
97 1−1.99iT−T2 1 - 1.99iT - 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

−9.675838861468718591496573939771, −8.966334859456534613954685250843, −8.049383399088252810255657580691, −7.26824637387688242128031184313, −6.29094244363363328176405738027, −5.26091075396337235046442407216, −4.81922782634146292338930599664, −3.70062011242145264556648357203, −2.23076976950991820046287219988, −0.983877278254931319822383704258, 1.75252778204371813879674385188, 2.98790083069800652164428781202, 3.96276922140315039872895976016, 4.91148072648213232272629511347, 5.62008423436526011214105108723, 7.06519947997700995730717498070, 7.49355768411657589788505623737, 8.535699455811223537970473250494, 8.932848126999100481305464682994, 9.823462040418878059960022066535

Graph of the ZZ-function along the critical line