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Properties

Label 2-2940-7.4-c1-0-20
Degree 22
Conductor 29402940
Sign −0.386+0.922i-0.386 + 0.922i
Analytic cond. 23.476023.4760
Root an. cond. 4.845204.84520
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)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (1 − 1.73i)11-s + 4·13-s − 0.999·15-s + (−1 + 1.73i)17-s + (−1 − 1.73i)19-s + (−2 − 3.46i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + 6·29-s + (1 − 1.73i)31-s + (−0.999 − 1.73i)33-s + (−5 − 8.66i)37-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.301 − 0.522i)11-s + 1.10·13-s − 0.258·15-s + (−0.242 + 0.420i)17-s + (−0.229 − 0.397i)19-s + (−0.417 − 0.722i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s + 1.11·29-s + (0.179 − 0.311i)31-s + (−0.174 − 0.301i)33-s + (−0.821 − 1.42i)37-s + ⋯

Functional equation

Λ(s)=(2940s/2 ΓC(s) L(s)=((−0.386+0.922i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2940s/2 ΓC(s+1/2) L(s)=((−0.386+0.922i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 29402940    =    22⋅3⋅5⋅722^{2} \cdot 3 \cdot 5 \cdot 7^{2}
Sign: −0.386+0.922i-0.386 + 0.922i
Analytic conductor: 23.476023.4760
Root analytic conductor: 4.845204.84520
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2940(361,⋅)\chi_{2940} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2940, ( :1/2), −0.386+0.922i)(2,\ 2940,\ (\ :1/2),\ -0.386 + 0.922i)

Particular Values

L(1)L(1) ≈\approx 1.7541552031.754155203
L(12)L(\frac12) ≈\approx 1.7541552031.754155203
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+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
5 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
7 1 1
good11 1+(−1+1.73i)T+(−5.5−9.52i)T2 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2}
13 1−4T+13T2 1 - 4T + 13T^{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+(1+1.73i)T+(−9.5+16.4i)T2 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2}
23 1+(2+3.46i)T+(−11.5+19.9i)T2 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2}
29 1−6T+29T2 1 - 6T + 29T^{2}
31 1+(−1+1.73i)T+(−15.5−26.8i)T2 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2}
37 1+(5+8.66i)T+(−18.5+32.0i)T2 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1−12T+43T2 1 - 12T + 43T^{2}
47 1+(−4−6.92i)T+(−23.5+40.7i)T2 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2}
53 1+(−26.5−45.8i)T2 1 + (-26.5 - 45.8i)T^{2}
59 1+(−4+6.92i)T+(−29.5−51.0i)T2 1 + (-4 + 6.92i)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+(−6+10.3i)T+(−33.5−58.0i)T2 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2}
71 1+10T+71T2 1 + 10T + 71T^{2}
73 1+(2−3.46i)T+(−36.5−63.2i)T2 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2}
79 1+(−39.5+68.4i)T2 1 + (-39.5 + 68.4i)T^{2}
83 1+12T+83T2 1 + 12T + 83T^{2}
89 1+(1+1.73i)T+(−44.5+77.0i)T2 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2}
97 1+8T+97T2 1 + 8T + 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

−8.655939034338345697155057699080, −7.907874656403813234861817784412, −6.97809132272929596079752408086, −6.26956590935373123519379768342, −5.62487466143993170523326065425, −4.43729587426430170656510290499, −3.77010574966112410097961211953, −2.76406401030701506625382780557, −1.64937687243616779230932029827, −0.56477112115496538302130946808, 1.34330332175008026168851722396, 2.54854771601251225415914081737, 3.50575859436075359254156725618, 4.13878536680193821875103740551, 5.03653109322354569477218550042, 5.97367861583938014292527882200, 6.76327145842183402637119278082, 7.46768965765711649883025369133, 8.498390417704503515117102502873, 8.757794150409406497292650805055

Graph of the ZZ-function along the critical line