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Properties

Label 2-325-13.12-c1-0-2
Degree 22
Conductor 325325
Sign −0.554−0.832i-0.554 - 0.832i
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
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  + i·2-s − 2·3-s + 4-s − 2i·6-s + 3i·8-s + 9-s + 2i·11-s − 2·12-s + (2 + 3i)13-s − 16-s + i·18-s + 6i·19-s − 2·22-s − 6·23-s − 6i·24-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.15·3-s + 0.5·4-s − 0.816i·6-s + 1.06i·8-s + 0.333·9-s + 0.603i·11-s − 0.577·12-s + (0.554 + 0.832i)13-s − 0.250·16-s + 0.235i·18-s + 1.37i·19-s − 0.426·22-s − 1.25·23-s − 1.22i·24-s + ⋯

Functional equation

Λ(s)=(325s/2 ΓC(s) L(s)=((−0.554−0.832i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(325s/2 ΓC(s+1/2) L(s)=((−0.554−0.832i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 325325    =    52⋅135^{2} \cdot 13
Sign: −0.554−0.832i-0.554 - 0.832i
Analytic conductor: 2.595132.59513
Root analytic conductor: 1.610941.61094
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ325(51,⋅)\chi_{325} (51, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 325, ( :1/2), −0.554−0.832i)(2,\ 325,\ (\ :1/2),\ -0.554 - 0.832i)

Particular Values

L(1)L(1) ≈\approx 0.436990+0.816524i0.436990 + 0.816524i
L(12)L(\frac12) ≈\approx 0.436990+0.816524i0.436990 + 0.816524i
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)
bad5 1 1
13 1+(−2−3i)T 1 + (-2 - 3i)T
good2 1−iT−2T2 1 - iT - 2T^{2}
3 1+2T+3T2 1 + 2T + 3T^{2}
7 1−7T2 1 - 7T^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
17 1+17T2 1 + 17T^{2}
19 1−6iT−19T2 1 - 6iT - 19T^{2}
23 1+6T+23T2 1 + 6T + 23T^{2}
29 1+6T+29T2 1 + 6T + 29T^{2}
31 1−6iT−31T2 1 - 6iT - 31T^{2}
37 1+6iT−37T2 1 + 6iT - 37T^{2}
41 1+8iT−41T2 1 + 8iT - 41T^{2}
43 1−6T+43T2 1 - 6T + 43T^{2}
47 1−8iT−47T2 1 - 8iT - 47T^{2}
53 1−12T+53T2 1 - 12T + 53T^{2}
59 1+2iT−59T2 1 + 2iT - 59T^{2}
61 1−6T+61T2 1 - 6T + 61T^{2}
67 1+12iT−67T2 1 + 12iT - 67T^{2}
71 1+2iT−71T2 1 + 2iT - 71T^{2}
73 1+6iT−73T2 1 + 6iT - 73T^{2}
79 1+79T2 1 + 79T^{2}
83 1+4iT−83T2 1 + 4iT - 83T^{2}
89 1−8iT−89T2 1 - 8iT - 89T^{2}
97 1−6iT−97T2 1 - 6iT - 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

−11.99296353276332942642765155921, −11.01254498328097680870653599007, −10.37743657056284865208759440962, −9.022312494215980019091934159935, −7.83680075981377800263635606552, −6.93064912686883245100387620651, −6.02407049889489475068721471980, −5.44260138555209788819174274601, −4.01076134625495095606087851020, −1.93515407303545731145842992812, 0.75437596206238912134373358332, 2.60985512727022564354621993941, 3.96864114407920061231982211201, 5.51158480258020253772074522027, 6.19833588257915813134827660667, 7.25335192544233749893786306720, 8.531257305818010614689127322682, 9.882562489060455031237515278807, 10.65534114512796297854884259577, 11.42992459237653641999388807895

Graph of the ZZ-function along the critical line