Properties

Label 2-20e2-400.141-c1-0-49
Degree $2$
Conductor $400$
Sign $-0.960 + 0.279i$
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  + (−0.690 − 1.23i)2-s + (1.40 − 2.76i)3-s + (−1.04 + 1.70i)4-s + (2.16 − 0.542i)5-s + (−4.38 + 0.169i)6-s + 0.132i·7-s + (2.82 + 0.115i)8-s + (−3.88 − 5.34i)9-s + (−2.16 − 2.30i)10-s + (−4.62 + 0.732i)11-s + (3.23 + 5.28i)12-s + (0.185 + 0.0293i)13-s + (0.163 − 0.0915i)14-s + (1.55 − 6.75i)15-s + (−1.80 − 3.56i)16-s + (0.766 − 2.35i)17-s + ⋯
L(s)  = 1  + (−0.488 − 0.872i)2-s + (0.812 − 1.59i)3-s + (−0.523 + 0.852i)4-s + (0.970 − 0.242i)5-s + (−1.78 + 0.0692i)6-s + 0.0501i·7-s + (0.999 + 0.0408i)8-s + (−1.29 − 1.78i)9-s + (−0.685 − 0.728i)10-s + (−1.39 + 0.220i)11-s + (0.933 + 1.52i)12-s + (0.0513 + 0.00813i)13-s + (0.0437 − 0.0244i)14-s + (0.401 − 1.74i)15-s + (−0.452 − 0.891i)16-s + (0.185 − 0.572i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.960 + 0.279i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.960 + 0.279i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197901 - 1.38647i\)
\(L(\frac12)\) \(\approx\) \(0.197901 - 1.38647i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.690 + 1.23i)T \)
5 \( 1 + (-2.16 + 0.542i)T \)
good3 \( 1 + (-1.40 + 2.76i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 - 0.132iT - 7T^{2} \)
11 \( 1 + (4.62 - 0.732i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (-0.185 - 0.0293i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.766 + 2.35i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.0229 + 0.0117i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-2.58 + 3.55i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-3.93 + 7.73i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (2.77 - 8.53i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.157 - 0.997i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.93 - 9.54i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.84 - 4.84i)T - 43iT^{2} \)
47 \( 1 + (-1.41 - 4.36i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.40 - 3.26i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (0.693 - 4.37i)T + (-56.1 - 18.2i)T^{2} \)
61 \( 1 + (1.05 + 6.68i)T + (-58.0 + 18.8i)T^{2} \)
67 \( 1 + (-3.47 + 1.76i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (-10.6 + 3.47i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.66 - 3.66i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (4.61 + 14.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-10.9 + 5.58i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (5.08 - 6.99i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.83 - 5.64i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.81449640007150931980267018794, −9.850413518087679817406084828541, −8.957069583777248267042060625630, −8.173277174868882047140303136056, −7.44437345170339828715899477076, −6.35589723155563534175939142128, −4.93374577631836966562026604239, −2.89235402762066060426814666853, −2.34957636409815367671259406018, −1.03905183588831366686937442894, 2.41510883815625211365649169782, 3.82661914968727479628927283212, 5.22655927546290294540145498282, 5.59135625189839230155072018884, 7.19342793285825605342617947364, 8.289558353642561951902506473503, 9.003931751318009543051586742242, 9.763706901106488977163660126644, 10.49481414390213046928929482067, 10.86809712456659073854783699023

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