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Properties

Label 2-416-13.12-c1-0-12
Degree 22
Conductor 416416
Sign −0.832−0.554i-0.832 - 0.554i
Analytic cond. 3.321773.32177
Root an. cond. 1.822571.82257
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3i·5-s + i·7-s + 6·9-s − 4i·11-s + (−2 + 3i)13-s + 9i·15-s − 5·17-s + 6i·19-s − 3i·21-s − 6·23-s − 4·25-s − 9·27-s − 4·29-s + 12i·33-s + ⋯
L(s)  = 1  − 1.73·3-s − 1.34i·5-s + 0.377i·7-s + 2·9-s − 1.20i·11-s + (−0.554 + 0.832i)13-s + 2.32i·15-s − 1.21·17-s + 1.37i·19-s − 0.654i·21-s − 1.25·23-s − 0.800·25-s − 1.73·27-s − 0.742·29-s + 2.08i·33-s + ⋯

Functional equation

Λ(s)=(416s/2 ΓC(s) L(s)=((−0.832−0.554i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(416s/2 ΓC(s+1/2) L(s)=((−0.832−0.554i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 416416    =    25⋅132^{5} \cdot 13
Sign: −0.832−0.554i-0.832 - 0.554i
Analytic conductor: 3.321773.32177
Root analytic conductor: 1.822571.82257
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ416(129,⋅)\chi_{416} (129, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 11
Selberg data: (2, 416, ( :1/2), −0.832−0.554i)(2,\ 416,\ (\ :1/2),\ -0.832 - 0.554i)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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
13 1+(2−3i)T 1 + (2 - 3i)T
good3 1+3T+3T2 1 + 3T + 3T^{2}
5 1+3iT−5T2 1 + 3iT - 5T^{2}
7 1−iT−7T2 1 - iT - 7T^{2}
11 1+4iT−11T2 1 + 4iT - 11T^{2}
17 1+5T+17T2 1 + 5T + 17T^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+4T+29T2 1 + 4T + 29T^{2}
31 1−31T2 1 - 31T^{2}
37 1−3iT−37T2 1 - 3iT - 37T^{2}
41 1−12iT−41T2 1 - 12iT - 41T^{2}
43 1+3T+43T2 1 + 3T + 43T^{2}
47 1+7iT−47T2 1 + 7iT - 47T^{2}
53 1+2T+53T2 1 + 2T + 53T^{2}
59 1−2iT−59T2 1 - 2iT - 59T^{2}
61 1+12T+61T2 1 + 12T + 61T^{2}
67 1−4iT−67T2 1 - 4iT - 67T^{2}
71 1+11iT−71T2 1 + 11iT - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1−10iT−83T2 1 - 10iT - 83T^{2}
89 1+6iT−89T2 1 + 6iT - 89T^{2}
97 1−97T2 1 - 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

−10.90058415489492309293663788560, −9.867705803055531930194665540861, −8.924066083756116351624652023821, −7.944595749674271891760273873132, −6.47010685473241215068343474096, −5.83759866425758622327451266892, −4.96700719456017829763534961263, −4.11737137647606119361774060872, −1.58958060048972467946282313599, 0, 2.31848070166838050919277286340, 4.12187527325456957596345917235, 5.10019492592589482081221636138, 6.20214531995203480907610222723, 7.00599394138295271063141268226, 7.46646046416840431551234300796, 9.428483445700316367073144700927, 10.45065652592070170228992426050, 10.72301841796029202465661168657

Graph of the ZZ-function along the critical line