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Properties

Label 2-48e2-48.11-c1-0-3
Degree 22
Conductor 23042304
Sign −0.884−0.465i-0.884 - 0.465i
Analytic cond. 18.397518.3975
Root an. cond. 4.289234.28923
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  + (−0.517 + 0.517i)5-s − 1.41·7-s + (2.73 + 2.73i)11-s + (−1.73 + 1.73i)13-s − 0.378i·17-s + (0.378 + 0.378i)19-s − 3.46i·23-s + 4.46i·25-s + (−4.76 − 4.76i)29-s + 0.656i·31-s + (0.732 − 0.732i)35-s + (4.46 + 4.46i)37-s − 10.1·41-s + (6.31 − 6.31i)43-s − 10.3·47-s + ⋯
L(s)  = 1  + (−0.231 + 0.231i)5-s − 0.534·7-s + (0.823 + 0.823i)11-s + (−0.480 + 0.480i)13-s − 0.0919i·17-s + (0.0869 + 0.0869i)19-s − 0.722i·23-s + 0.892i·25-s + (−0.883 − 0.883i)29-s + 0.117i·31-s + (0.123 − 0.123i)35-s + (0.733 + 0.733i)37-s − 1.58·41-s + (0.962 − 0.962i)43-s − 1.51·47-s + ⋯

Functional equation

Λ(s)=(2304s/2 ΓC(s) L(s)=((−0.884−0.465i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2304s/2 ΓC(s+1/2) L(s)=((−0.884−0.465i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 23042304    =    28⋅322^{8} \cdot 3^{2}
Sign: −0.884−0.465i-0.884 - 0.465i
Analytic conductor: 18.397518.3975
Root analytic conductor: 4.289234.28923
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2304(575,⋅)\chi_{2304} (575, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2304, ( :1/2), −0.884−0.465i)(2,\ 2304,\ (\ :1/2),\ -0.884 - 0.465i)

Particular Values

L(1)L(1) ≈\approx 0.62927440660.6292744066
L(12)L(\frac12) ≈\approx 0.62927440660.6292744066
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 1
3 1 1
good5 1+(0.517−0.517i)T−5iT2 1 + (0.517 - 0.517i)T - 5iT^{2}
7 1+1.41T+7T2 1 + 1.41T + 7T^{2}
11 1+(−2.73−2.73i)T+11iT2 1 + (-2.73 - 2.73i)T + 11iT^{2}
13 1+(1.73−1.73i)T−13iT2 1 + (1.73 - 1.73i)T - 13iT^{2}
17 1+0.378iT−17T2 1 + 0.378iT - 17T^{2}
19 1+(−0.378−0.378i)T+19iT2 1 + (-0.378 - 0.378i)T + 19iT^{2}
23 1+3.46iT−23T2 1 + 3.46iT - 23T^{2}
29 1+(4.76+4.76i)T+29iT2 1 + (4.76 + 4.76i)T + 29iT^{2}
31 1−0.656iT−31T2 1 - 0.656iT - 31T^{2}
37 1+(−4.46−4.46i)T+37iT2 1 + (-4.46 - 4.46i)T + 37iT^{2}
41 1+10.1T+41T2 1 + 10.1T + 41T^{2}
43 1+(−6.31+6.31i)T−43iT2 1 + (-6.31 + 6.31i)T - 43iT^{2}
47 1+10.3T+47T2 1 + 10.3T + 47T^{2}
53 1+(4.00−4.00i)T−53iT2 1 + (4.00 - 4.00i)T - 53iT^{2}
59 1+(−4.92−4.92i)T+59iT2 1 + (-4.92 - 4.92i)T + 59iT^{2}
61 1+(3−3i)T−61iT2 1 + (3 - 3i)T - 61iT^{2}
67 1+(−1.03−1.03i)T+67iT2 1 + (-1.03 - 1.03i)T + 67iT^{2}
71 1+14iT−71T2 1 + 14iT - 71T^{2}
73 1−8.92iT−73T2 1 - 8.92iT - 73T^{2}
79 1−11.9iT−79T2 1 - 11.9iT - 79T^{2}
83 1+(7.26−7.26i)T−83iT2 1 + (7.26 - 7.26i)T - 83iT^{2}
89 1+13.2T+89T2 1 + 13.2T + 89T^{2}
97 1−2.39T+97T2 1 - 2.39T + 97T^{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

−9.604080862383583134921183367213, −8.603878613899114838513122452513, −7.69386680106043844543957921936, −6.91606811661523084025781578247, −6.46080866506200006372944905177, −5.36749977618084969723722601126, −4.42351641922495016069318373094, −3.69562016608128407554883561247, −2.64164727623675172840482702375, −1.53410337432609390943695285265, 0.21331046711891688065956724056, 1.57455437724618792499478798722, 2.99401130541178182733400973168, 3.64844705079875142732115220974, 4.65485810074121922988096468052, 5.59307229474604362273031334417, 6.33307738483322999405816153656, 7.10523842458888787623099753976, 8.004989536395056544161532757979, 8.643695980955433512122659216453

Graph of the ZZ-function along the critical line