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Properties

Label 2-20e2-100.27-c1-0-12
Degree $2$
Conductor $400$
Sign $0.166 + 0.986i$
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  + (1.00 − 0.510i)3-s + (−0.844 − 2.07i)5-s + (0.941 − 0.941i)7-s + (−1.02 + 1.40i)9-s + (−3.06 − 4.21i)11-s + (6.49 − 1.02i)13-s + (−1.90 − 1.64i)15-s + (0.833 − 1.63i)17-s + (−0.753 − 2.31i)19-s + (0.462 − 1.42i)21-s + (−7.19 − 1.13i)23-s + (−3.57 + 3.49i)25-s + (−0.832 + 5.25i)27-s + (5.17 + 1.68i)29-s + (9.74 − 3.16i)31-s + ⋯
L(s)  = 1  + (0.578 − 0.294i)3-s + (−0.377 − 0.925i)5-s + (0.355 − 0.355i)7-s + (−0.340 + 0.468i)9-s + (−0.923 − 1.27i)11-s + (1.80 − 0.285i)13-s + (−0.491 − 0.424i)15-s + (0.202 − 0.396i)17-s + (−0.172 − 0.532i)19-s + (0.100 − 0.310i)21-s + (−1.49 − 0.237i)23-s + (−0.714 + 0.699i)25-s + (−0.160 + 1.01i)27-s + (0.960 + 0.312i)29-s + (1.75 − 0.568i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.166 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.166 + 0.986i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.166 + 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14121 - 0.964781i\)
\(L(\frac12)\) \(\approx\) \(1.14121 - 0.964781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.844 + 2.07i)T \)
good3 \( 1 + (-1.00 + 0.510i)T + (1.76 - 2.42i)T^{2} \)
7 \( 1 + (-0.941 + 0.941i)T - 7iT^{2} \)
11 \( 1 + (3.06 + 4.21i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-6.49 + 1.02i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-0.833 + 1.63i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (0.753 + 2.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (7.19 + 1.13i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-5.17 - 1.68i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-9.74 + 3.16i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.379 - 2.39i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (0.485 + 0.352i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (4.84 + 4.84i)T + 43iT^{2} \)
47 \( 1 + (-2.93 - 5.76i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.53 - 3.01i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (-8.77 - 6.37i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.27 + 1.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-6.68 - 3.40i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (6.56 + 2.13i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.502 + 3.16i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.00 - 6.15i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.97 - 3.88i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (-4.09 - 5.62i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (10.6 - 5.43i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.10693913930391976111620427120, −10.29240438025098710666100555641, −8.782315556027848150135559244526, −8.289994147534370351612788213459, −7.84054271784236296974874688111, −6.20090126688751058150485090809, −5.24135289177966620230810083770, −4.01093500101411006383004169201, −2.77231083276902230327075024780, −0.986543015205870279688164044720, 2.15007080109218486053946406375, 3.39071143523792914522036816181, 4.30784414769923473172158944670, 5.88707225825502114605773998777, 6.73289277506808207320222591954, 8.154977403662827567029634935286, 8.377796409889473933852549013246, 9.909438734427349613467723551206, 10.34649776029918457262944266467, 11.54944386655734467911804153344

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