Properties

Label 2-20e2-400.181-c1-0-52
Degree $2$
Conductor $400$
Sign $-0.245 + 0.969i$
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  + (1.19 − 0.749i)2-s + (0.337 − 2.12i)3-s + (0.876 − 1.79i)4-s + (2.01 + 0.968i)5-s + (−1.19 − 2.80i)6-s − 1.96i·7-s + (−0.295 − 2.81i)8-s + (−1.56 − 0.508i)9-s + (3.14 − 0.349i)10-s + (−2.64 + 5.19i)11-s + (−3.53 − 2.47i)12-s + (1.39 + 2.74i)13-s + (−1.47 − 2.35i)14-s + (2.74 − 3.96i)15-s + (−2.46 − 3.15i)16-s + (1.68 + 1.22i)17-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.194 − 1.22i)3-s + (0.438 − 0.898i)4-s + (0.901 + 0.433i)5-s + (−0.486 − 1.14i)6-s − 0.742i·7-s + (−0.104 − 0.994i)8-s + (−0.522 − 0.169i)9-s + (0.993 − 0.110i)10-s + (−0.797 + 1.56i)11-s + (−1.01 − 0.713i)12-s + (0.387 + 0.761i)13-s + (−0.393 − 0.629i)14-s + (0.707 − 1.02i)15-s + (−0.615 − 0.788i)16-s + (0.408 + 0.296i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.245 + 0.969i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.245 + 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57260 - 2.02087i\)
\(L(\frac12)\) \(\approx\) \(1.57260 - 2.02087i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.19 + 0.749i)T \)
5 \( 1 + (-2.01 - 0.968i)T \)
good3 \( 1 + (-0.337 + 2.12i)T + (-2.85 - 0.927i)T^{2} \)
7 \( 1 + 1.96iT - 7T^{2} \)
11 \( 1 + (2.64 - 5.19i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (-1.39 - 2.74i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.68 - 1.22i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.93 - 0.306i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (5.58 - 1.81i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (0.527 - 3.33i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (2.31 + 1.68i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-7.34 + 3.74i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (10.6 + 3.46i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (0.295 + 0.295i)T + 43iT^{2} \)
47 \( 1 + (4.33 - 3.14i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (7.92 + 1.25i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-12.2 + 6.24i)T + (34.6 - 47.7i)T^{2} \)
61 \( 1 + (3.43 + 1.74i)T + (35.8 + 49.3i)T^{2} \)
67 \( 1 + (-8.77 + 1.38i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-3.51 - 4.83i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-13.3 + 4.34i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (6.44 - 4.68i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-8.72 + 1.38i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (-3.32 + 1.07i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (13.9 - 10.1i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.10109341687755507830391814283, −10.19821934705354817043257986763, −9.601874267258732394543822195316, −7.86114486783586065016697165607, −6.97458954821542296249447843230, −6.40388414293476574420990145980, −5.17632082725414584596892254082, −3.89331923840140666180064022290, −2.27045127197807614446989435348, −1.65713375183366359003185944992, 2.62385907195632977590415731859, 3.60233074601983388738261215105, 4.92416931177780150787256634236, 5.59275091986308443655615718385, 6.28591247142319143376486792963, 8.179912993759653905299164073304, 8.586531245973604859766154015018, 9.748925149783301967008330539093, 10.58940395827157273402790281074, 11.52524540154742510822937958570

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