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Properties

Label 2-15e2-15.8-c1-0-3
Degree 22
Conductor 225225
Sign −0.876+0.481i-0.876 + 0.481i
Analytic cond. 1.796631.79663
Root an. cond. 1.340381.34038
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  + (−1.73 − 1.73i)2-s + 3.99i·4-s + (1.22 − 1.22i)7-s + (3.46 − 3.46i)8-s − 4.24i·11-s + (−3.67 − 3.67i)13-s − 4.24·14-s − 3.99·16-s + (1.73 + 1.73i)17-s − 5i·19-s + (−7.34 + 7.34i)22-s + (−1.73 + 1.73i)23-s + 12.7i·26-s + (4.89 + 4.89i)28-s − 4.24·29-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)2-s + 1.99i·4-s + (0.462 − 0.462i)7-s + (1.22 − 1.22i)8-s − 1.27i·11-s + (−1.01 − 1.01i)13-s − 1.13·14-s − 0.999·16-s + (0.420 + 0.420i)17-s − 1.14i·19-s + (−1.56 + 1.56i)22-s + (−0.361 + 0.361i)23-s + 2.49i·26-s + (0.925 + 0.925i)28-s − 0.787·29-s + ⋯

Functional equation

Λ(s)=(225s/2 ΓC(s) L(s)=((−0.876+0.481i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(225s/2 ΓC(s+1/2) L(s)=((−0.876+0.481i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 225225    =    32⋅523^{2} \cdot 5^{2}
Sign: −0.876+0.481i-0.876 + 0.481i
Analytic conductor: 1.796631.79663
Root analytic conductor: 1.340381.34038
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ225(143,⋅)\chi_{225} (143, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 225, ( :1/2), −0.876+0.481i)(2,\ 225,\ (\ :1/2),\ -0.876 + 0.481i)

Particular Values

L(1)L(1) ≈\approx 0.142229−0.554215i0.142229 - 0.554215i
L(12)L(\frac12) ≈\approx 0.142229−0.554215i0.142229 - 0.554215i
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)
bad3 1 1
5 1 1
good2 1+(1.73+1.73i)T+2iT2 1 + (1.73 + 1.73i)T + 2iT^{2}
7 1+(−1.22+1.22i)T−7iT2 1 + (-1.22 + 1.22i)T - 7iT^{2}
11 1+4.24iT−11T2 1 + 4.24iT - 11T^{2}
13 1+(3.67+3.67i)T+13iT2 1 + (3.67 + 3.67i)T + 13iT^{2}
17 1+(−1.73−1.73i)T+17iT2 1 + (-1.73 - 1.73i)T + 17iT^{2}
19 1+5iT−19T2 1 + 5iT - 19T^{2}
23 1+(1.73−1.73i)T−23iT2 1 + (1.73 - 1.73i)T - 23iT^{2}
29 1+4.24T+29T2 1 + 4.24T + 29T^{2}
31 1−T+31T2 1 - T + 31T^{2}
37 1+(−2.44+2.44i)T−37iT2 1 + (-2.44 + 2.44i)T - 37iT^{2}
41 1+8.48iT−41T2 1 + 8.48iT - 41T^{2}
43 1+(1.22+1.22i)T+43iT2 1 + (1.22 + 1.22i)T + 43iT^{2}
47 1+(−5.19−5.19i)T+47iT2 1 + (-5.19 - 5.19i)T + 47iT^{2}
53 1+(−6.92+6.92i)T−53iT2 1 + (-6.92 + 6.92i)T - 53iT^{2}
59 1−12.7T+59T2 1 - 12.7T + 59T^{2}
61 1+7T+61T2 1 + 7T + 61T^{2}
67 1+(−3.67+3.67i)T−67iT2 1 + (-3.67 + 3.67i)T - 67iT^{2}
71 1−8.48iT−71T2 1 - 8.48iT - 71T^{2}
73 1+(−2.44−2.44i)T+73iT2 1 + (-2.44 - 2.44i)T + 73iT^{2}
79 1−2iT−79T2 1 - 2iT - 79T^{2}
83 1+(1.73−1.73i)T−83iT2 1 + (1.73 - 1.73i)T - 83iT^{2}
89 1+8.48T+89T2 1 + 8.48T + 89T^{2}
97 1+(8.57−8.57i)T−97iT2 1 + (8.57 - 8.57i)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

−11.47590850967784301914375074342, −10.82830197752965162340494667071, −10.06358246872476802783446505034, −9.046415633833492609780552262527, −8.137131401640558919117113142037, −7.34616539281897897653365323572, −5.48793683036491656874411270155, −3.73322246618618312885169621328, −2.51179118645682904468238432214, −0.71881077159523051674139860729, 1.91646681515026274660386773448, 4.59764290092272401539857748534, 5.73695267404867297466864619600, 6.94932558777755585588789359843, 7.62843142538659275611409059202, 8.609706231969304663220920502309, 9.663364362850317436610656346382, 10.08642708235202175042730428718, 11.62457466386676957517898233086, 12.46124525936878305766155436051

Graph of the ZZ-function along the critical line