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Properties

Label 2-2548-13.5-c0-0-1
Degree 22
Conductor 25482548
Sign 0.881+0.471i0.881 + 0.471i
Analytic cond. 1.271611.27161
Root an. cond. 1.127661.12766
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + (0.707 − 0.707i)5-s + 1.00·9-s + (−0.707 + 0.707i)13-s + (1.00 − 1.00i)15-s − 1.41i·17-s + (0.707 − 0.707i)19-s − i·23-s − 29-s + (−0.707 + 0.707i)31-s + (1 + i)37-s + (−1.00 + 1.00i)39-s + i·43-s + (0.707 − 0.707i)45-s + (0.707 + 0.707i)47-s + ⋯
L(s)  = 1  + 1.41·3-s + (0.707 − 0.707i)5-s + 1.00·9-s + (−0.707 + 0.707i)13-s + (1.00 − 1.00i)15-s − 1.41i·17-s + (0.707 − 0.707i)19-s − i·23-s − 29-s + (−0.707 + 0.707i)31-s + (1 + i)37-s + (−1.00 + 1.00i)39-s + i·43-s + (0.707 − 0.707i)45-s + (0.707 + 0.707i)47-s + ⋯

Functional equation

Λ(s)=(2548s/2 ΓC(s) L(s)=((0.881+0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2548s/2 ΓC(s) L(s)=((0.881+0.471i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 25482548    =    22⋅72⋅132^{2} \cdot 7^{2} \cdot 13
Sign: 0.881+0.471i0.881 + 0.471i
Analytic conductor: 1.271611.27161
Root analytic conductor: 1.127661.12766
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2548(785,⋅)\chi_{2548} (785, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2548, ( :0), 0.881+0.471i)(2,\ 2548,\ (\ :0),\ 0.881 + 0.471i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.1171698722.117169872
L(12)L(\frac12) ≈\approx 2.1171698722.117169872
L(1)L(1) 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+(0.707−0.707i)T 1 + (0.707 - 0.707i)T
good3 1−1.41T+T2 1 - 1.41T + T^{2}
5 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
11 1+iT2 1 + iT^{2}
17 1+1.41iT−T2 1 + 1.41iT - T^{2}
19 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
23 1+iT−T2 1 + iT - T^{2}
29 1+T+T2 1 + T + T^{2}
31 1+(0.707−0.707i)T−iT2 1 + (0.707 - 0.707i)T - iT^{2}
37 1+(−1−i)T+iT2 1 + (-1 - i)T + iT^{2}
41 1−iT2 1 - iT^{2}
43 1−iT−T2 1 - iT - T^{2}
47 1+(−0.707−0.707i)T+iT2 1 + (-0.707 - 0.707i)T + iT^{2}
53 1+T+T2 1 + T + T^{2}
59 1+(−1.41−1.41i)T+iT2 1 + (-1.41 - 1.41i)T + iT^{2}
61 1−1.41T+T2 1 - 1.41T + T^{2}
67 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
71 1+(1−i)T−iT2 1 + (1 - i)T - iT^{2}
73 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
79 1+T+T2 1 + T + T^{2}
83 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{2}
89 1+(0.707+0.707i)T+iT2 1 + (0.707 + 0.707i)T + iT^{2}
97 1+(−0.707+0.707i)T−iT2 1 + (-0.707 + 0.707i)T - iT^{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

−9.036449901136071275297841152563, −8.540076325582319403604566526151, −7.43292022374580725509501548629, −7.11203608918565685806731698020, −5.84463365836190152423035512698, −4.93849624922076652841012025673, −4.28816444566749275919479489651, −2.99989622638371912573610358766, −2.47852947228702082011571428318, −1.37124271307593974350886651711, 1.81678024406073891736278018308, 2.41530868358777968719648294845, 3.43487267265167992252872095127, 3.92624054978879261851738218880, 5.45227604154539325389210406615, 5.96000962558609064104567865223, 7.13260258721512848625763794384, 7.71436284554181993969117046501, 8.293709979583084834011616115125, 9.259595317374064530416293645582

Graph of the ZZ-function along the critical line