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Properties

Label 2-20e2-25.14-c1-0-13
Degree $2$
Conductor $400$
Sign $-0.962 + 0.271i$
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  + (−0.166 − 0.0542i)3-s + (−1.44 − 1.70i)5-s − 1.27i·7-s + (−2.40 − 1.74i)9-s + (−3.52 + 2.56i)11-s + (−2.51 + 3.46i)13-s + (0.148 + 0.363i)15-s + (−0.0930 + 0.0302i)17-s + (−0.103 − 0.317i)19-s + (−0.0690 + 0.212i)21-s + (−3.71 − 5.10i)23-s + (−0.819 + 4.93i)25-s + (0.616 + 0.847i)27-s + (1.44 − 4.44i)29-s + (−2.48 − 7.64i)31-s + ⋯
L(s)  = 1  + (−0.0964 − 0.0313i)3-s + (−0.646 − 0.762i)5-s − 0.481i·7-s + (−0.800 − 0.581i)9-s + (−1.06 + 0.773i)11-s + (−0.698 + 0.961i)13-s + (0.0384 + 0.0937i)15-s + (−0.0225 + 0.00733i)17-s + (−0.0236 − 0.0728i)19-s + (−0.0150 + 0.0464i)21-s + (−0.773 − 1.06i)23-s + (−0.163 + 0.986i)25-s + (0.118 + 0.163i)27-s + (0.268 − 0.826i)29-s + (−0.446 − 1.37i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.962 + 0.271i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.962 + 0.271i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0498373 - 0.359964i\)
\(L(\frac12)\) \(\approx\) \(0.0498373 - 0.359964i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.44 + 1.70i)T \)
good3 \( 1 + (0.166 + 0.0542i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 + 1.27iT - 7T^{2} \)
11 \( 1 + (3.52 - 2.56i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.51 - 3.46i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.0930 - 0.0302i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.103 + 0.317i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (3.71 + 5.10i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.44 + 4.44i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.48 + 7.64i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.41 + 3.32i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.24 + 6.71i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.2iT - 43T^{2} \)
47 \( 1 + (-11.1 - 3.61i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.841 + 0.273i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.15 + 4.47i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.35 + 0.985i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.11 - 0.361i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (-0.728 + 2.24i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.945 + 1.30i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-2.51 + 7.73i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (7.05 - 2.29i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-1.38 + 1.00i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-6.28 - 2.04i)T + (78.4 + 57.0i)T^{2} \)
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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

−10.96011166120257922001377756504, −9.877914803985535403604838107288, −9.036654909863124151752201388656, −8.014887516374636057496023105571, −7.27953791374959251387419729435, −6.02915584847696442298898839067, −4.80621635657296329097407357107, −4.02382134342446618167467765912, −2.36267710349571459800754450942, −0.22304490212798419200610696221, 2.58424071161841139487648751258, 3.37961946612037204464286537868, 5.12977169191182676560820656098, 5.77827186251629571071173983487, 7.17731144227382141864461759737, 8.015738656358509967989382665490, 8.693117417379457024048361143142, 10.25448803051118028161747931804, 10.67972427915401954660668336045, 11.65439440282316048012615450173

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