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Properties

Label 2-20e2-80.69-c1-0-7
Degree $2$
Conductor $400$
Sign $-0.896 - 0.443i$
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  + (0.139 + 1.40i)2-s + (2.32 + 2.32i)3-s + (−1.96 + 0.393i)4-s + (−2.94 + 3.59i)6-s + 0.982·7-s + (−0.828 − 2.70i)8-s + 7.82i·9-s + (−1.62 − 1.62i)11-s + (−5.47 − 3.64i)12-s + (0.690 + 0.690i)13-s + (0.137 + 1.38i)14-s + (3.68 − 1.54i)16-s − 2.19i·17-s + (−11.0 + 1.09i)18-s + (−1.92 + 1.92i)19-s + ⋯
L(s)  = 1  + (0.0989 + 0.995i)2-s + (1.34 + 1.34i)3-s + (−0.980 + 0.196i)4-s + (−1.20 + 1.46i)6-s + 0.371·7-s + (−0.292 − 0.956i)8-s + 2.60i·9-s + (−0.490 − 0.490i)11-s + (−1.58 − 1.05i)12-s + (0.191 + 0.191i)13-s + (0.0367 + 0.369i)14-s + (0.922 − 0.386i)16-s − 0.532i·17-s + (−2.59 + 0.258i)18-s + (−0.441 + 0.441i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.896 - 0.443i$
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: \((2,\ 400,\ (\ :1/2),\ -0.896 - 0.443i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.447741 + 1.91280i\)
\(L(\frac12)\) \(\approx\) \(0.447741 + 1.91280i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.139 - 1.40i)T \)
5 \( 1 \)
good3 \( 1 + (-2.32 - 2.32i)T + 3iT^{2} \)
7 \( 1 - 0.982T + 7T^{2} \)
11 \( 1 + (1.62 + 1.62i)T + 11iT^{2} \)
13 \( 1 + (-0.690 - 0.690i)T + 13iT^{2} \)
17 \( 1 + 2.19iT - 17T^{2} \)
19 \( 1 + (1.92 - 1.92i)T - 19iT^{2} \)
23 \( 1 - 2.01T + 23T^{2} \)
29 \( 1 + (-5.27 + 5.27i)T - 29iT^{2} \)
31 \( 1 - 0.435T + 31T^{2} \)
37 \( 1 + (-5.79 + 5.79i)T - 37iT^{2} \)
41 \( 1 - 3.93iT - 41T^{2} \)
43 \( 1 + (0.507 - 0.507i)T - 43iT^{2} \)
47 \( 1 + 9.21iT - 47T^{2} \)
53 \( 1 + (-6.29 + 6.29i)T - 53iT^{2} \)
59 \( 1 + (-5.67 - 5.67i)T + 59iT^{2} \)
61 \( 1 + (3.60 - 3.60i)T - 61iT^{2} \)
67 \( 1 + (-4.53 - 4.53i)T + 67iT^{2} \)
71 \( 1 - 10.3iT - 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-0.683 - 0.683i)T + 83iT^{2} \)
89 \( 1 + 5.44iT - 89T^{2} \)
97 \( 1 - 5.54iT - 97T^{2} \)
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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

−11.45805451687670551026132071446, −10.28909457503380188028976787787, −9.688388545339141507638401058903, −8.621875937797809978016619309372, −8.298862452551231135219183286116, −7.27029198557220438068859065324, −5.71483701624042539832805740666, −4.70339323075221379503104311561, −3.91730011891570106502630220781, −2.73123487344230048738795948618, 1.26462813226226590954668949149, 2.38458702012862381680430444008, 3.28510595182556914862460461841, 4.64834790601992890587661938676, 6.25362601110466260899184988961, 7.44917598566597923669525325821, 8.301369751161565064200563197330, 8.893489369512204495934831378910, 9.909160406583979342815354729510, 10.99393876229592439624639788086

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