Properties

Label 2-20e2-16.5-c1-0-15
Degree $2$
Conductor $400$
Sign $0.522 - 0.852i$
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  + (1.41 + 0.0554i)2-s + (−0.488 + 0.488i)3-s + (1.99 + 0.156i)4-s + (−0.717 + 0.663i)6-s + 4.71i·7-s + (2.80 + 0.331i)8-s + 2.52i·9-s + (−3.91 − 3.91i)11-s + (−1.05 + 0.897i)12-s + (−0.0878 + 0.0878i)13-s + (−0.261 + 6.66i)14-s + (3.95 + 0.624i)16-s + 4.67·17-s + (−0.139 + 3.56i)18-s + (1.81 − 1.81i)19-s + ⋯
L(s)  = 1  + (0.999 + 0.0391i)2-s + (−0.282 + 0.282i)3-s + (0.996 + 0.0783i)4-s + (−0.292 + 0.270i)6-s + 1.78i·7-s + (0.993 + 0.117i)8-s + 0.840i·9-s + (−1.17 − 1.17i)11-s + (−0.303 + 0.259i)12-s + (−0.0243 + 0.0243i)13-s + (−0.0698 + 1.78i)14-s + (0.987 + 0.156i)16-s + 1.13·17-s + (−0.0329 + 0.840i)18-s + (0.415 − 0.415i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.522 - 0.852i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97265 + 1.10509i\)
\(L(\frac12)\) \(\approx\) \(1.97265 + 1.10509i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.0554i)T \)
5 \( 1 \)
good3 \( 1 + (0.488 - 0.488i)T - 3iT^{2} \)
7 \( 1 - 4.71iT - 7T^{2} \)
11 \( 1 + (3.91 + 3.91i)T + 11iT^{2} \)
13 \( 1 + (0.0878 - 0.0878i)T - 13iT^{2} \)
17 \( 1 - 4.67T + 17T^{2} \)
19 \( 1 + (-1.81 + 1.81i)T - 19iT^{2} \)
23 \( 1 + 1.63iT - 23T^{2} \)
29 \( 1 + (-3.26 + 3.26i)T - 29iT^{2} \)
31 \( 1 + 2.12T + 31T^{2} \)
37 \( 1 + (-3.97 - 3.97i)T + 37iT^{2} \)
41 \( 1 + 8.25iT - 41T^{2} \)
43 \( 1 + (-2.27 - 2.27i)T + 43iT^{2} \)
47 \( 1 + 4.06T + 47T^{2} \)
53 \( 1 + (5.03 + 5.03i)T + 53iT^{2} \)
59 \( 1 + (5.16 + 5.16i)T + 59iT^{2} \)
61 \( 1 + (-7.12 + 7.12i)T - 61iT^{2} \)
67 \( 1 + (7.49 - 7.49i)T - 67iT^{2} \)
71 \( 1 - 4.54iT - 71T^{2} \)
73 \( 1 + 8.30iT - 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 + (1.16 - 1.16i)T - 83iT^{2} \)
89 \( 1 - 3.24iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−11.50883343296982819400470788530, −10.84154366083045637902448438529, −9.817371248761831898107691011732, −8.416810933250544639899397042103, −7.78032701637979917578257010985, −6.18099127677852910887213373203, −5.47466631411292742930739837316, −4.94672352972963696679649504469, −3.18095048296227479223226002243, −2.34301571494514906660227187418, 1.28237290090231208808413980119, 3.16037431254161597579064794721, 4.18256199538220922390439864635, 5.18099954821067766251614559986, 6.35031043832860360365417010405, 7.41850607550410874989676745040, 7.64640473098342257001074061679, 9.787943159740122465936600150568, 10.33512095926126988810389055626, 11.22144237081698602786599112515

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