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Properties

Label 2-2548-91.88-c1-0-0
Degree 22
Conductor 25482548
Sign −0.895+0.444i-0.895 + 0.444i
Analytic cond. 20.345820.3458
Root an. cond. 4.510644.51064
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  − 1.22·3-s + (−2.34 + 1.35i)5-s − 1.48·9-s − 1.33i·11-s + (2.87 + 2.17i)13-s + (2.87 − 1.66i)15-s + (1.24 + 2.16i)17-s + 4.59i·19-s + (0.104 − 0.180i)23-s + (1.15 − 1.99i)25-s + 5.51·27-s + (−0.639 − 1.10i)29-s + (4.89 + 2.82i)31-s + 1.63i·33-s + (−2.07 − 1.19i)37-s + ⋯
L(s)  = 1  − 0.710·3-s + (−1.04 + 0.604i)5-s − 0.495·9-s − 0.401i·11-s + (0.798 + 0.601i)13-s + (0.743 − 0.429i)15-s + (0.302 + 0.524i)17-s + 1.05i·19-s + (0.0217 − 0.0376i)23-s + (0.230 − 0.399i)25-s + 1.06·27-s + (−0.118 − 0.205i)29-s + (0.878 + 0.507i)31-s + 0.285i·33-s + (−0.340 − 0.196i)37-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((−0.895+0.444i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s+1/2) L(s)=((−0.895+0.444i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: −0.895+0.444i-0.895 + 0.444i
Analytic conductor: 20.345820.3458
Root analytic conductor: 4.510644.51064
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ2548(361,⋅)\chi_{2548} (361, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :1/2), −0.895+0.444i)(2,\ 2548,\ (\ :1/2),\ -0.895 + 0.444i)

Particular Values

L(1)L(1) ≈\approx 0.14428216190.1442821619
L(12)L(\frac12) ≈\approx 0.14428216190.1442821619
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
7 1 1
13 1+(−2.87−2.17i)T 1 + (-2.87 - 2.17i)T
good3 1+1.22T+3T2 1 + 1.22T + 3T^{2}
5 1+(2.34−1.35i)T+(2.5−4.33i)T2 1 + (2.34 - 1.35i)T + (2.5 - 4.33i)T^{2}
11 1+1.33iT−11T2 1 + 1.33iT - 11T^{2}
17 1+(−1.24−2.16i)T+(−8.5+14.7i)T2 1 + (-1.24 - 2.16i)T + (-8.5 + 14.7i)T^{2}
19 1−4.59iT−19T2 1 - 4.59iT - 19T^{2}
23 1+(−0.104+0.180i)T+(−11.5−19.9i)T2 1 + (-0.104 + 0.180i)T + (-11.5 - 19.9i)T^{2}
29 1+(0.639+1.10i)T+(−14.5+25.1i)T2 1 + (0.639 + 1.10i)T + (-14.5 + 25.1i)T^{2}
31 1+(−4.89−2.82i)T+(15.5+26.8i)T2 1 + (-4.89 - 2.82i)T + (15.5 + 26.8i)T^{2}
37 1+(2.07+1.19i)T+(18.5+32.0i)T2 1 + (2.07 + 1.19i)T + (18.5 + 32.0i)T^{2}
41 1+(2.55−1.47i)T+(20.5−35.5i)T2 1 + (2.55 - 1.47i)T + (20.5 - 35.5i)T^{2}
43 1+(0.348−0.602i)T+(−21.5−37.2i)T2 1 + (0.348 - 0.602i)T + (-21.5 - 37.2i)T^{2}
47 1+(2.11−1.22i)T+(23.5−40.7i)T2 1 + (2.11 - 1.22i)T + (23.5 - 40.7i)T^{2}
53 1+(1.09−1.89i)T+(−26.5−45.8i)T2 1 + (1.09 - 1.89i)T + (-26.5 - 45.8i)T^{2}
59 1+(4.87−2.81i)T+(29.5−51.0i)T2 1 + (4.87 - 2.81i)T + (29.5 - 51.0i)T^{2}
61 1+12.3T+61T2 1 + 12.3T + 61T^{2}
67 1+14.1iT−67T2 1 + 14.1iT - 67T^{2}
71 1+(1.70+0.984i)T+(35.5+61.4i)T2 1 + (1.70 + 0.984i)T + (35.5 + 61.4i)T^{2}
73 1+(−10.7−6.21i)T+(36.5+63.2i)T2 1 + (-10.7 - 6.21i)T + (36.5 + 63.2i)T^{2}
79 1+(−4.16−7.21i)T+(−39.5+68.4i)T2 1 + (-4.16 - 7.21i)T + (-39.5 + 68.4i)T^{2}
83 1+2.83iT−83T2 1 + 2.83iT - 83T^{2}
89 1+(10.3+5.99i)T+(44.5+77.0i)T2 1 + (10.3 + 5.99i)T + (44.5 + 77.0i)T^{2}
97 1+(14.1+8.17i)T+(48.5+84.0i)T2 1 + (14.1 + 8.17i)T + (48.5 + 84.0i)T^{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

−9.284006693116414788431042624209, −8.307115920139795618853190785542, −7.981584807507733617817166982889, −6.90171416319939063586124228240, −6.25310014419668925169272790667, −5.62529946087733080130184534502, −4.53025784412907327827133000467, −3.67673491582578100693322549375, −3.00584300755632241633451782287, −1.43743059282071135201590354740, 0.06576927156018149211050524156, 1.04621279135781085722294237544, 2.71936550248965195477191374573, 3.66460887683461883173863864011, 4.65559364373401351699080665863, 5.19105254793894779425565218883, 6.10259391483382254001648723097, 6.90860427100306668651545486375, 7.79887828377255553760771077880, 8.404396160514558559469077126624

Graph of the ZZ-function along the critical line