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Properties

Label 2-20e2-25.21-c1-0-0
Degree $2$
Conductor $400$
Sign $-0.452 - 0.891i$
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  + (−1.09 − 0.792i)3-s + (−2.15 + 0.587i)5-s + 0.833·7-s + (−0.365 − 1.12i)9-s + (−0.257 + 0.792i)11-s + (1.41 + 4.34i)13-s + (2.81 + 1.06i)15-s + (−4.41 + 3.20i)17-s + (−7.00 + 5.08i)19-s + (−0.909 − 0.660i)21-s + (−1.09 + 3.35i)23-s + (4.30 − 2.53i)25-s + (−1.74 + 5.36i)27-s + (−2.64 − 1.92i)29-s + (4.85 − 3.52i)31-s + ⋯
L(s)  = 1  + (−0.629 − 0.457i)3-s + (−0.964 + 0.262i)5-s + 0.314·7-s + (−0.121 − 0.374i)9-s + (−0.0776 + 0.238i)11-s + (0.391 + 1.20i)13-s + (0.727 + 0.275i)15-s + (−1.06 + 0.777i)17-s + (−1.60 + 1.16i)19-s + (−0.198 − 0.144i)21-s + (−0.227 + 0.700i)23-s + (0.861 − 0.507i)25-s + (−0.335 + 1.03i)27-s + (−0.491 − 0.356i)29-s + (0.872 − 0.633i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.452 - 0.891i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.452 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.212075 + 0.345266i\)
\(L(\frac12)\) \(\approx\) \(0.212075 + 0.345266i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.15 - 0.587i)T \)
good3 \( 1 + (1.09 + 0.792i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 - 0.833T + 7T^{2} \)
11 \( 1 + (0.257 - 0.792i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.41 - 4.34i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.41 - 3.20i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (7.00 - 5.08i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.09 - 3.35i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (2.64 + 1.92i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.85 + 3.52i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.26 - 6.95i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.576 - 1.77i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.63T + 43T^{2} \)
47 \( 1 + (0.674 + 0.489i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.19 + 3.77i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.18 + 12.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.81 + 5.59i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-1.21 + 0.881i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (1.91 + 1.38i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.02 - 3.16i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.18 - 3.03i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.97 - 7.25i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.16 - 6.66i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (8.97 + 6.51i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.34389161172044032381743260883, −11.18083488837919964016744327668, −9.830558385400482118279897711985, −8.600346475915028261840707387949, −7.898488452228242646406845302378, −6.61142103219684949756022805038, −6.24264548738581102950009371352, −4.55522031247897903042816024119, −3.74580815180084944187574598497, −1.82279669891208126504957026322, 0.27977580754852042001591506174, 2.70973418060711279423613479319, 4.30116058726315211800872192891, 4.89649044579597701961611330796, 6.08348773232015131297236855116, 7.29362424459393286434916396823, 8.317504604737165295081207664030, 8.947027218684630615436629500548, 10.54178822135789460191536580387, 10.90752202690063551544291923771

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