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Properties

Label 2-325-1.1-c1-0-15
Degree 22
Conductor 325325
Sign 11
Analytic cond. 2.595132.59513
Root an. cond. 1.610941.61094
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·2-s + 1.41·3-s + 3.82·4-s + 3.41·6-s − 4.82·7-s + 4.41·8-s − 0.999·9-s + 3.41·11-s + 5.41·12-s + 13-s − 11.6·14-s + 2.99·16-s − 0.828·17-s − 2.41·18-s + 0.585·19-s − 6.82·21-s + 8.24·22-s − 1.41·23-s + 6.24·24-s + 2.41·26-s − 5.65·27-s − 18.4·28-s − 5.65·29-s + 1.75·31-s − 1.58·32-s + 4.82·33-s − 1.99·34-s + ⋯
L(s)  = 1  + 1.70·2-s + 0.816·3-s + 1.91·4-s + 1.39·6-s − 1.82·7-s + 1.56·8-s − 0.333·9-s + 1.02·11-s + 1.56·12-s + 0.277·13-s − 3.11·14-s + 0.749·16-s − 0.200·17-s − 0.569·18-s + 0.134·19-s − 1.49·21-s + 1.75·22-s − 0.294·23-s + 1.27·24-s + 0.473·26-s − 1.08·27-s − 3.49·28-s − 1.05·29-s + 0.315·31-s − 0.280·32-s + 0.840·33-s − 0.342·34-s + ⋯

Functional equation

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

Invariants

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

Particular Values

L(1)L(1) ≈\approx 3.5154624783.515462478
L(12)L(\frac12) ≈\approx 3.5154624783.515462478
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−T 1 - T
good2 1−2.41T+2T2 1 - 2.41T + 2T^{2}
3 1−1.41T+3T2 1 - 1.41T + 3T^{2}
7 1+4.82T+7T2 1 + 4.82T + 7T^{2}
11 1−3.41T+11T2 1 - 3.41T + 11T^{2}
17 1+0.828T+17T2 1 + 0.828T + 17T^{2}
19 1−0.585T+19T2 1 - 0.585T + 19T^{2}
23 1+1.41T+23T2 1 + 1.41T + 23T^{2}
29 1+5.65T+29T2 1 + 5.65T + 29T^{2}
31 1−1.75T+31T2 1 - 1.75T + 31T^{2}
37 1−8.48T+37T2 1 - 8.48T + 37T^{2}
41 1+3.17T+41T2 1 + 3.17T + 41T^{2}
43 1−11.0T+43T2 1 - 11.0T + 43T^{2}
47 1−4.82T+47T2 1 - 4.82T + 47T^{2}
53 1+2.48T+53T2 1 + 2.48T + 53T^{2}
59 1−1.75T+59T2 1 - 1.75T + 59T^{2}
61 1+8T+61T2 1 + 8T + 61T^{2}
67 1−2T+67T2 1 - 2T + 67T^{2}
71 1−11.8T+71T2 1 - 11.8T + 71T^{2}
73 1+8.48T+73T2 1 + 8.48T + 73T^{2}
79 1+8.48T+79T2 1 + 8.48T + 79T^{2}
83 1−3.17T+83T2 1 - 3.17T + 83T^{2}
89 1−6T+89T2 1 - 6T + 89T^{2}
97 1−7.65T+97T2 1 - 7.65T + 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

−11.98851720669363499114006441604, −11.03695148106214761386622810969, −9.637457415543295880970012135246, −8.965366051204050460810662430924, −7.40532028527185204058298052249, −6.36564015065318793199579584516, −5.81638356123165706209115181429, −4.11233001423733721128704766639, −3.43407149252504548147344905218, −2.52103154495935654149200801156, 2.52103154495935654149200801156, 3.43407149252504548147344905218, 4.11233001423733721128704766639, 5.81638356123165706209115181429, 6.36564015065318793199579584516, 7.40532028527185204058298052249, 8.965366051204050460810662430924, 9.637457415543295880970012135246, 11.03695148106214761386622810969, 11.98851720669363499114006441604

Graph of the ZZ-function along the critical line