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Properties

Label 2-20e2-100.3-c1-0-11
Degree $2$
Conductor $400$
Sign $0.871 + 0.491i$
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  + (3.10 − 0.491i)3-s + (−0.657 − 2.13i)5-s + (1.48 + 1.48i)7-s + (6.53 − 2.12i)9-s + (−1.78 − 0.581i)11-s + (−3.02 + 5.94i)13-s + (−3.08 − 6.30i)15-s + (0.378 − 2.39i)17-s + (2.43 − 1.76i)19-s + (5.32 + 3.87i)21-s + (−2.55 − 5.01i)23-s + (−4.13 + 2.80i)25-s + (10.8 − 5.52i)27-s + (−2.79 + 3.84i)29-s + (−1.62 − 2.23i)31-s + ⋯
L(s)  = 1  + (1.79 − 0.283i)3-s + (−0.293 − 0.955i)5-s + (0.560 + 0.560i)7-s + (2.17 − 0.707i)9-s + (−0.539 − 0.175i)11-s + (−0.839 + 1.64i)13-s + (−0.797 − 1.62i)15-s + (0.0918 − 0.580i)17-s + (0.558 − 0.405i)19-s + (1.16 + 0.844i)21-s + (−0.532 − 1.04i)23-s + (−0.827 + 0.561i)25-s + (2.08 − 1.06i)27-s + (−0.518 + 0.714i)29-s + (−0.292 − 0.402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.491i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.871 + 0.491i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.871 + 0.491i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.24443 - 0.589303i\)
\(L(\frac12)\) \(\approx\) \(2.24443 - 0.589303i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.657 + 2.13i)T \)
good3 \( 1 + (-3.10 + 0.491i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (-1.48 - 1.48i)T + 7iT^{2} \)
11 \( 1 + (1.78 + 0.581i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (3.02 - 5.94i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.378 + 2.39i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-2.43 + 1.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.55 + 5.01i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (2.79 - 3.84i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.62 + 2.23i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (-3.13 - 1.59i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-2.32 - 7.16i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (8.80 - 8.80i)T - 43iT^{2} \)
47 \( 1 + (1.05 + 6.67i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-1.14 - 7.19i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.807 - 2.48i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (0.283 - 0.872i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-8.27 - 1.31i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-5.33 + 7.34i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.04 + 0.533i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (4.61 + 3.35i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.967 - 6.10i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (-0.0514 - 0.0167i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (15.5 - 2.46i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.45517231548941177905969598028, −9.729279920091337773444352607675, −9.241022942187258257196855181249, −8.433753259706510656903215601952, −7.81882109371125117179055789224, −6.85279554092464072080908078480, −5.02838812744264890078257521313, −4.19401880672338223503275304852, −2.75030460986395550175792267168, −1.73243946095558468212981856419, 2.12871606959058214303178677278, 3.20704289152575047325506664855, 3.95179204255411209115145212608, 5.40713159128213028606857665497, 7.24967299691993452920409072565, 7.75572094120358736025438497061, 8.277813490127438511531252643596, 9.760477683597856958990877276888, 10.15352761268504266870877532159, 11.04013369539051928130089649022

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