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Properties

Label 2-20e2-400.187-c1-0-31
Degree $2$
Conductor $400$
Sign $0.728 + 0.685i$
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  + (0.921 − 1.07i)2-s + (−0.121 − 0.375i)3-s + (−0.303 − 1.97i)4-s + (0.445 + 2.19i)5-s + (−0.515 − 0.214i)6-s + (3.43 + 3.43i)7-s + (−2.40 − 1.49i)8-s + (2.30 − 1.67i)9-s + (2.76 + 1.54i)10-s + (0.833 − 0.132i)11-s + (−0.704 + 0.354i)12-s + (0.687 + 0.946i)13-s + (6.84 − 0.522i)14-s + (0.768 − 0.434i)15-s + (−3.81 + 1.19i)16-s + (−3.37 + 1.72i)17-s + ⋯
L(s)  = 1  + (0.651 − 0.758i)2-s + (−0.0704 − 0.216i)3-s + (−0.151 − 0.988i)4-s + (0.199 + 0.979i)5-s + (−0.210 − 0.0876i)6-s + (1.29 + 1.29i)7-s + (−0.848 − 0.528i)8-s + (0.767 − 0.557i)9-s + (0.873 + 0.487i)10-s + (0.251 − 0.0398i)11-s + (−0.203 + 0.102i)12-s + (0.190 + 0.262i)13-s + (1.83 − 0.139i)14-s + (0.198 − 0.112i)15-s + (−0.954 + 0.299i)16-s + (−0.819 + 0.417i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.728 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.728 + 0.685i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.728 + 0.685i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.98449 - 0.787217i\)
\(L(\frac12)\) \(\approx\) \(1.98449 - 0.787217i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.921 + 1.07i)T \)
5 \( 1 + (-0.445 - 2.19i)T \)
good3 \( 1 + (0.121 + 0.375i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-3.43 - 3.43i)T + 7iT^{2} \)
11 \( 1 + (-0.833 + 0.132i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (-0.687 - 0.946i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.37 - 1.72i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (3.40 + 6.67i)T + (-11.1 + 15.3i)T^{2} \)
23 \( 1 + (-7.01 + 1.11i)T + (21.8 - 7.10i)T^{2} \)
29 \( 1 + (0.828 - 1.62i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (3.04 + 0.989i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (2.56 + 3.53i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.24 - 4.47i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.51iT - 43T^{2} \)
47 \( 1 + (4.56 + 2.32i)T + (27.6 + 38.0i)T^{2} \)
53 \( 1 + (-2.04 - 6.30i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.54 + 0.403i)T + (56.1 + 18.2i)T^{2} \)
61 \( 1 + (13.6 - 2.16i)T + (58.0 - 18.8i)T^{2} \)
67 \( 1 + (-13.6 - 4.43i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.89 - 5.82i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.795 + 5.02i)T + (-69.4 + 22.5i)T^{2} \)
79 \( 1 + (2.07 + 6.38i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.0485 + 0.149i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-0.947 - 0.688i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.12 - 10.0i)T + (-57.0 - 78.4i)T^{2} \)
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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

−11.07709249240186455189256902870, −10.82562956114671118094311236834, −9.316402633595713162074778543050, −8.779263905731621286101541249670, −7.08778170324548962155744472382, −6.32616799207018381263785870205, −5.20618110660669339071617194201, −4.18062782023992165295884715287, −2.69194998097279675483510719417, −1.75545680194399278836019300375, 1.59583826105232493899930423938, 3.92253024601913466074534632607, 4.60747201174469956442753785071, 5.28115550153996872105700265102, 6.70054343498348894873784594326, 7.72592161399080572427807266502, 8.258878027668688918989303655509, 9.404852664279662644900582720291, 10.64136976240915290320750181562, 11.38469897585387049044159059643

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