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Properties

Label 2-20e2-80.69-c1-0-21
Degree $2$
Conductor $400$
Sign $0.254 + 0.967i$
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  + (0.388 + 1.35i)2-s + (−1.03 − 1.03i)3-s + (−1.69 + 1.05i)4-s + (1.01 − 1.81i)6-s − 1.49·7-s + (−2.09 − 1.89i)8-s − 0.836i·9-s + (0.423 + 0.423i)11-s + (2.86 + 0.666i)12-s + (−1.85 − 1.85i)13-s + (−0.581 − 2.03i)14-s + (1.76 − 3.58i)16-s − 6.50i·17-s + (1.13 − 0.325i)18-s + (1.75 − 1.75i)19-s + ⋯
L(s)  = 1  + (0.274 + 0.961i)2-s + (−0.600 − 0.600i)3-s + (−0.849 + 0.528i)4-s + (0.412 − 0.742i)6-s − 0.565·7-s + (−0.741 − 0.671i)8-s − 0.278i·9-s + (0.127 + 0.127i)11-s + (0.827 + 0.192i)12-s + (−0.515 − 0.515i)13-s + (−0.155 − 0.543i)14-s + (0.441 − 0.897i)16-s − 1.57i·17-s + (0.268 − 0.0766i)18-s + (0.403 − 0.403i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.254 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.254 + 0.967i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.254 + 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.487364 - 0.375844i\)
\(L(\frac12)\) \(\approx\) \(0.487364 - 0.375844i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.388 - 1.35i)T \)
5 \( 1 \)
good3 \( 1 + (1.03 + 1.03i)T + 3iT^{2} \)
7 \( 1 + 1.49T + 7T^{2} \)
11 \( 1 + (-0.423 - 0.423i)T + 11iT^{2} \)
13 \( 1 + (1.85 + 1.85i)T + 13iT^{2} \)
17 \( 1 + 6.50iT - 17T^{2} \)
19 \( 1 + (-1.75 + 1.75i)T - 19iT^{2} \)
23 \( 1 + 7.19T + 23T^{2} \)
29 \( 1 + (-6.57 + 6.57i)T - 29iT^{2} \)
31 \( 1 + 6.75T + 31T^{2} \)
37 \( 1 + (-1.95 + 1.95i)T - 37iT^{2} \)
41 \( 1 - 7.70iT - 41T^{2} \)
43 \( 1 + (-6.13 + 6.13i)T - 43iT^{2} \)
47 \( 1 - 6.65iT - 47T^{2} \)
53 \( 1 + (5.29 - 5.29i)T - 53iT^{2} \)
59 \( 1 + (5.91 + 5.91i)T + 59iT^{2} \)
61 \( 1 + (1.43 - 1.43i)T - 61iT^{2} \)
67 \( 1 + (-6.35 - 6.35i)T + 67iT^{2} \)
71 \( 1 + 4.08iT - 71T^{2} \)
73 \( 1 - 2.43T + 73T^{2} \)
79 \( 1 + 11.6T + 79T^{2} \)
83 \( 1 + (-2.81 - 2.81i)T + 83iT^{2} \)
89 \( 1 - 10.5iT - 89T^{2} \)
97 \( 1 + 18.1iT - 97T^{2} \)
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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

−11.41191323306444302741383886926, −9.813771985416086047111856520447, −9.319581402401218657765021356173, −7.917612680397924405923581447327, −7.19198824351512699365712254443, −6.34146323888757232588537126548, −5.57222467882703797359639058172, −4.40597857906141272248289478715, −2.97923580501916827134472677849, −0.39008922320020435385773784119, 1.91961911700979297909045518016, 3.51758888813181010888687561482, 4.41401080368300183379709686599, 5.48967882322751839267843409780, 6.34672471069798121443277761498, 7.979768911893488226592748470905, 9.079125545175910190931479984339, 10.05957501029036179256329801282, 10.49149258411608042215903927602, 11.39612993390105230987106900989

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