Properties

Label 2-20e2-400.141-c1-0-36
Degree $2$
Conductor $400$
Sign $0.738 + 0.674i$
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  + (−1.38 − 0.274i)2-s + (0.652 − 1.28i)3-s + (1.84 + 0.762i)4-s + (2.03 + 0.925i)5-s + (−1.25 + 1.59i)6-s − 1.67i·7-s + (−2.35 − 1.56i)8-s + (0.549 + 0.756i)9-s + (−2.56 − 1.84i)10-s + (−0.376 + 0.0595i)11-s + (2.18 − 1.86i)12-s + (5.88 + 0.931i)13-s + (−0.461 + 2.33i)14-s + (2.51 − 2.00i)15-s + (2.83 + 2.81i)16-s + (1.49 − 4.61i)17-s + ⋯
L(s)  = 1  + (−0.980 − 0.194i)2-s + (0.376 − 0.739i)3-s + (0.924 + 0.381i)4-s + (0.910 + 0.413i)5-s + (−0.513 + 0.651i)6-s − 0.634i·7-s + (−0.832 − 0.553i)8-s + (0.183 + 0.252i)9-s + (−0.812 − 0.582i)10-s + (−0.113 + 0.0179i)11-s + (0.629 − 0.539i)12-s + (1.63 + 0.258i)13-s + (−0.123 + 0.622i)14-s + (0.648 − 0.517i)15-s + (0.709 + 0.704i)16-s + (0.363 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.738 + 0.674i$
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: \((2,\ 400,\ (\ :1/2),\ 0.738 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16449 - 0.451830i\)
\(L(\frac12)\) \(\approx\) \(1.16449 - 0.451830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.274i)T \)
5 \( 1 + (-2.03 - 0.925i)T \)
good3 \( 1 + (-0.652 + 1.28i)T + (-1.76 - 2.42i)T^{2} \)
7 \( 1 + 1.67iT - 7T^{2} \)
11 \( 1 + (0.376 - 0.0595i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (-5.88 - 0.931i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.49 + 4.61i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.97 - 3.04i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (4.03 - 5.55i)T + (-7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.964 + 1.89i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (-1.74 + 5.36i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.233 + 1.47i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (0.668 + 0.920i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.99 + 4.99i)T - 43iT^{2} \)
47 \( 1 + (1.93 + 5.95i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (12.4 + 6.36i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (1.85 - 11.7i)T + (-56.1 - 18.2i)T^{2} \)
61 \( 1 + (1.35 + 8.53i)T + (-58.0 + 18.8i)T^{2} \)
67 \( 1 + (1.76 - 0.899i)T + (39.3 - 54.2i)T^{2} \)
71 \( 1 + (8.22 - 2.67i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (6.36 - 8.75i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.52 - 7.76i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.69 + 3.40i)T + (48.7 - 67.1i)T^{2} \)
89 \( 1 + (5.91 - 8.14i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.60 - 11.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.88486375731419341543183710536, −10.26722666602974077459395316353, −9.375966916207770249233176871493, −8.344877211088822112885022366537, −7.56676996921378397398129370441, −6.68030887576964758598544437867, −5.86989491681701499254861739218, −3.79635018230778072435922817940, −2.36024884104081847245133977344, −1.37022151740766313829222294787, 1.51982425756154666603351131522, 2.96268628713763779274132109056, 4.52180465791094065793349586652, 6.07041019102626814873341749570, 6.35988419936744665865540640586, 8.255783602259327160341248889436, 8.720296386038211759479508047185, 9.373875476812349957525825299505, 10.50207503574615520290866747567, 10.70651471679535301377075035379

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