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Properties

Label 2-936-104.83-c1-0-56
Degree 22
Conductor 936936
Sign 0.289+0.957i0.289 + 0.957i
Analytic cond. 7.473997.47399
Root an. cond. 2.733862.73386
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  + (1 − i)2-s − 2i·4-s + (2.54 + 2.54i)5-s + (2.54 − 2.54i)7-s + (−2 − 2i)8-s + 5.09·10-s + (−1 − i)11-s + (−2.54 − 2.54i)13-s − 5.09i·14-s − 4·16-s + 3i·17-s + (2 − 2i)19-s + (5.09 − 5.09i)20-s − 2·22-s + 5.09·23-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·4-s + (1.14 + 1.14i)5-s + (0.963 − 0.963i)7-s + (−0.707 − 0.707i)8-s + 1.61·10-s + (−0.301 − 0.301i)11-s + (−0.707 − 0.707i)13-s − 1.36i·14-s − 16-s + 0.727i·17-s + (0.458 − 0.458i)19-s + (1.14 − 1.14i)20-s − 0.426·22-s + 1.06·23-s + ⋯

Functional equation

Λ(s)=(936s/2 ΓC(s) L(s)=((0.289+0.957i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(936s/2 ΓC(s+1/2) L(s)=((0.289+0.957i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.289 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 936936    =    23⋅32⋅132^{3} \cdot 3^{2} \cdot 13
Sign: 0.289+0.957i0.289 + 0.957i
Analytic conductor: 7.473997.47399
Root analytic conductor: 2.733862.73386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ936(811,⋅)\chi_{936} (811, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 936, ( :1/2), 0.289+0.957i)(2,\ 936,\ (\ :1/2),\ 0.289 + 0.957i)

Particular Values

L(1)L(1) ≈\approx 2.33683−1.73406i2.33683 - 1.73406i
L(12)L(\frac12) ≈\approx 2.33683−1.73406i2.33683 - 1.73406i
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+i)T 1 + (-1 + i)T
3 1 1
13 1+(2.54+2.54i)T 1 + (2.54 + 2.54i)T
good5 1+(−2.54−2.54i)T+5iT2 1 + (-2.54 - 2.54i)T + 5iT^{2}
7 1+(−2.54+2.54i)T−7iT2 1 + (-2.54 + 2.54i)T - 7iT^{2}
11 1+(1+i)T+11iT2 1 + (1 + i)T + 11iT^{2}
17 1−3iT−17T2 1 - 3iT - 17T^{2}
19 1+(−2+2i)T−19iT2 1 + (-2 + 2i)T - 19iT^{2}
23 1−5.09T+23T2 1 - 5.09T + 23T^{2}
29 1+5.09iT−29T2 1 + 5.09iT - 29T^{2}
31 1+(−5.09−5.09i)T+31iT2 1 + (-5.09 - 5.09i)T + 31iT^{2}
37 1+(−2.54+2.54i)T−37iT2 1 + (-2.54 + 2.54i)T - 37iT^{2}
41 1+(6−6i)T−41iT2 1 + (6 - 6i)T - 41iT^{2}
43 1+iT−43T2 1 + iT - 43T^{2}
47 1+(2.54−2.54i)T−47iT2 1 + (2.54 - 2.54i)T - 47iT^{2}
53 1−5.09iT−53T2 1 - 5.09iT - 53T^{2}
59 1+(8+8i)T+59iT2 1 + (8 + 8i)T + 59iT^{2}
61 1−61T2 1 - 61T^{2}
67 1+(−3+3i)T−67iT2 1 + (-3 + 3i)T - 67iT^{2}
71 1+(−7.64−7.64i)T+71iT2 1 + (-7.64 - 7.64i)T + 71iT^{2}
73 1+(6+6i)T+73iT2 1 + (6 + 6i)T + 73iT^{2}
79 1−5.09iT−79T2 1 - 5.09iT - 79T^{2}
83 1+(5−5i)T−83iT2 1 + (5 - 5i)T - 83iT^{2}
89 1+(−2−2i)T+89iT2 1 + (-2 - 2i)T + 89iT^{2}
97 1+(7−7i)T−97iT2 1 + (7 - 7i)T - 97iT^{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

−10.12966773014694335943590476120, −9.557353044162453250250627797339, −8.142846438640600119715787099806, −7.11184405366992576233665452012, −6.33738263526808625532998095193, −5.37097271656677812342122588842, −4.60704097818432399880515948968, −3.26888033588431467602705817501, −2.48765451709671433120912480656, −1.25558492530026760144186621694, 1.74750822922777253876146994724, 2.74035986022406899856085015498, 4.52823294538890658032763869976, 5.11849672562400602854232721136, 5.55827311868999934229248287371, 6.67370875336032050667200054176, 7.68209547190769230461329148660, 8.619512165212573419724487081709, 9.122256044269451672589827430802, 9.937949912302899723180063404412

Graph of the ZZ-function along the critical line