Properties

Label 2-20e2-25.14-c1-0-3
Degree $2$
Conductor $400$
Sign $0.400 - 0.916i$
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  + (−1.54 − 0.501i)3-s + (2.03 + 0.923i)5-s + 2.51i·7-s + (−0.292 − 0.212i)9-s + (0.362 − 0.263i)11-s + (−3.75 + 5.17i)13-s + (−2.68 − 2.44i)15-s + (3.87 − 1.26i)17-s + (0.114 + 0.352i)19-s + (1.26 − 3.88i)21-s + (4.27 + 5.88i)23-s + (3.29 + 3.76i)25-s + (3.20 + 4.41i)27-s + (−1.82 + 5.62i)29-s + (−0.492 − 1.51i)31-s + ⋯
L(s)  = 1  + (−0.891 − 0.289i)3-s + (0.910 + 0.413i)5-s + 0.951i·7-s + (−0.0975 − 0.0708i)9-s + (0.109 − 0.0794i)11-s + (−1.04 + 1.43i)13-s + (−0.692 − 0.632i)15-s + (0.940 − 0.305i)17-s + (0.0262 + 0.0808i)19-s + (0.275 − 0.848i)21-s + (0.890 + 1.22i)23-s + (0.658 + 0.752i)25-s + (0.617 + 0.850i)27-s + (−0.339 + 1.04i)29-s + (−0.0885 − 0.272i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.400 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.400 - 0.916i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.400 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.866696 + 0.567079i\)
\(L(\frac12)\) \(\approx\) \(0.866696 + 0.567079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 0.923i)T \)
good3 \( 1 + (1.54 + 0.501i)T + (2.42 + 1.76i)T^{2} \)
7 \( 1 - 2.51iT - 7T^{2} \)
11 \( 1 + (-0.362 + 0.263i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.75 - 5.17i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.87 + 1.26i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-0.114 - 0.352i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-4.27 - 5.88i)T + (-7.10 + 21.8i)T^{2} \)
29 \( 1 + (1.82 - 5.62i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.492 + 1.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.485 + 0.668i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (2.38 + 1.73i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.31iT - 43T^{2} \)
47 \( 1 + (-11.2 - 3.64i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (11.7 + 3.81i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (10.2 + 7.43i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.23 + 5.25i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (2.88 - 0.938i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.44 - 4.43i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.41 + 3.32i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.22 + 13.0i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.37 - 1.42i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + (-6.08 + 4.41i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.67 - 0.870i)T + (78.4 + 57.0i)T^{2} \)
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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

−11.58440663264078315252278385931, −10.68412413624441044913176967963, −9.414245509581827223687984379440, −9.135430304785981403474325276347, −7.40542446092845383768300192783, −6.59554034643617708029034886057, −5.67416270136907201552430930643, −5.04651535162598974093020694918, −3.10114415281628972379673906238, −1.74605996657310348042962359549, 0.78287172936226206935973136512, 2.75887089816938500867726059699, 4.48335657424092342316901621525, 5.32327250041976618145855632894, 6.08575735291064309556526788824, 7.30190882285547495866080349202, 8.307769239264314145279484628518, 9.612876145728300571858707167574, 10.38606508502172128992102792510, 10.75463136112669104268647825415

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