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Properties

Label 2-20e2-100.27-c1-0-4
Degree $2$
Conductor $400$
Sign $-0.0673 - 0.997i$
Analytic cond. $3.19401$
Root an. cond. $1.78718$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.97 + 1.51i)3-s + (1.69 + 1.45i)5-s + (1.80 − 1.80i)7-s + (4.80 − 6.61i)9-s + (2.46 + 3.39i)11-s + (0.731 − 0.115i)13-s + (−7.26 − 1.76i)15-s + (−3.09 + 6.08i)17-s + (0.285 + 0.877i)19-s + (−2.63 + 8.10i)21-s + (−3.66 − 0.579i)23-s + (0.753 + 4.94i)25-s + (−2.70 + 17.0i)27-s + (−2.48 − 0.808i)29-s + (5.85 − 1.90i)31-s + ⋯
L(s)  = 1  + (−1.71 + 0.876i)3-s + (0.758 + 0.651i)5-s + (0.681 − 0.681i)7-s + (1.60 − 2.20i)9-s + (0.743 + 1.02i)11-s + (0.202 − 0.0321i)13-s + (−1.87 − 0.456i)15-s + (−0.751 + 1.47i)17-s + (0.0653 + 0.201i)19-s + (−0.574 + 1.76i)21-s + (−0.763 − 0.120i)23-s + (0.150 + 0.988i)25-s + (−0.520 + 3.28i)27-s + (−0.462 − 0.150i)29-s + (1.05 − 0.341i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0673 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.0673 - 0.997i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.0673 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640902 + 0.685645i\)
\(L(\frac12)\) \(\approx\) \(0.640902 + 0.685645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.69 - 1.45i)T \)
good3 \( 1 + (2.97 - 1.51i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-1.80 + 1.80i)T - 7iT^{2} \)
11 \( 1 + (-2.46 - 3.39i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.731 + 0.115i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (3.09 - 6.08i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-0.285 - 0.877i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.66 + 0.579i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (2.48 + 0.808i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.85 + 1.90i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.443 + 2.79i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (-2.09 - 1.52i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-1.85 - 1.85i)T + 43iT^{2} \)
47 \( 1 + (-3.13 - 6.15i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.786 - 1.54i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-1.10 - 0.803i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (6.13 - 4.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (8.25 + 4.20i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (-5.26 - 1.71i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.380 + 2.40i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-1.89 + 5.82i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.10 - 2.16i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (5.88 + 8.09i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-7.27 + 3.70i)T + (57.0 - 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.25624733760191228408402063806, −10.60349567382711583631959929928, −10.09078369223113986770752214339, −9.202202339946898203303851038783, −7.47357624474988366148228234696, −6.39561029723586226346977303354, −5.93627030067282796501048034115, −4.58812780978632242961456764465, −4.00634676897108344556292119752, −1.56773512573196279765147817836, 0.845275558661012419551379080891, 2.09955271388581715263552075923, 4.63768812113616285565132745776, 5.41767974574921160181859832470, 6.11080705320614540937154628422, 6.95194931495880922631541077222, 8.234260705095642197344623692431, 9.190195327222392721681192214553, 10.39246067915401064771416873574, 11.46611459429946702837601117964

Graph of the $Z$-function along the critical line