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Properties

Label 2-20e2-100.3-c1-0-14
Degree $2$
Conductor $400$
Sign $-0.957 + 0.289i$
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  + (0.608 − 0.0963i)3-s + (−2.10 − 0.757i)5-s + (−1.38 − 1.38i)7-s + (−2.49 + 0.809i)9-s + (−5.27 − 1.71i)11-s + (−0.110 + 0.216i)13-s + (−1.35 − 0.257i)15-s + (−0.974 + 6.15i)17-s + (3.29 − 2.39i)19-s + (−0.973 − 0.707i)21-s + (−3.85 − 7.57i)23-s + (3.85 + 3.18i)25-s + (−3.08 + 1.57i)27-s + (2.97 − 4.08i)29-s + (1.96 + 2.70i)31-s + ⋯
L(s)  = 1  + (0.351 − 0.0555i)3-s + (−0.940 − 0.338i)5-s + (−0.522 − 0.522i)7-s + (−0.830 + 0.269i)9-s + (−1.59 − 0.516i)11-s + (−0.0305 + 0.0599i)13-s + (−0.349 − 0.0665i)15-s + (−0.236 + 1.49i)17-s + (0.754 − 0.548i)19-s + (−0.212 − 0.154i)21-s + (−0.804 − 1.57i)23-s + (0.770 + 0.637i)25-s + (−0.593 + 0.302i)27-s + (0.551 − 0.759i)29-s + (0.352 + 0.485i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0471105 - 0.318722i\)
\(L(\frac12)\) \(\approx\) \(0.0471105 - 0.318722i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.10 + 0.757i)T \)
good3 \( 1 + (-0.608 + 0.0963i)T + (2.85 - 0.927i)T^{2} \)
7 \( 1 + (1.38 + 1.38i)T + 7iT^{2} \)
11 \( 1 + (5.27 + 1.71i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.110 - 0.216i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (0.974 - 6.15i)T + (-16.1 - 5.25i)T^{2} \)
19 \( 1 + (-3.29 + 2.39i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (3.85 + 7.57i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-2.97 + 4.08i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.96 - 2.70i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.87 + 4.01i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (-2.53 - 7.80i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + (-2.80 + 2.80i)T - 43iT^{2} \)
47 \( 1 + (1.06 + 6.69i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (0.431 + 2.72i)T + (-50.4 + 16.3i)T^{2} \)
59 \( 1 + (-0.496 - 1.52i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.00 + 3.09i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (9.09 + 1.44i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (9.23 - 12.7i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (4.14 - 2.11i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-9.67 - 7.02i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-1.63 + 10.3i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (5.96 + 1.93i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.71 + 0.746i)T + (92.2 - 29.9i)T^{2} \)
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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

−10.75866629114451481187684471999, −10.18615704245679336537198221772, −8.590138534487895913517397775747, −8.296458289166734664041331250954, −7.32604114336618160455162740315, −6.06094519493574200627271702788, −4.89210922166714799258816583196, −3.68923286537484730388663343569, −2.62430197090690536467992349002, −0.18731124063916062937961606593, 2.67277920012769228409405049825, 3.35014387940698256172301500654, 4.91115261708241300009568412056, 5.88853130661575711748364777688, 7.33997417421872578810788508477, 7.83727839080197664291386899125, 8.951898575698000785309735244741, 9.805824200832754376491803328737, 10.81372498414627853980868408350, 11.83919763486372730397386727051

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