↑

Properties

Label 2-20e2-16.5-c1-0-33
Degree $2$
Conductor $400$
Sign $-0.901 + 0.431i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.257 − 1.39i)2-s + (1.66 − 1.66i)3-s + (−1.86 − 0.715i)4-s + (−1.88 − 2.74i)6-s − 2.89i·7-s + (−1.47 + 2.41i)8-s − 2.53i·9-s + (1.84 + 1.84i)11-s + (−4.29 + 1.91i)12-s + (3.08 − 3.08i)13-s + (−4.02 − 0.744i)14-s + (2.97 + 2.67i)16-s − 7.29·17-s + (−3.52 − 0.652i)18-s + (−1.23 + 1.23i)19-s + ⋯
L(s)  = 1  + (0.181 − 0.983i)2-s + (0.960 − 0.960i)3-s + (−0.933 − 0.357i)4-s + (−0.769 − 1.11i)6-s − 1.09i·7-s + (−0.521 + 0.853i)8-s − 0.845i·9-s + (0.556 + 0.556i)11-s + (−1.24 + 0.553i)12-s + (0.854 − 0.854i)13-s + (−1.07 − 0.198i)14-s + (0.744 + 0.667i)16-s − 1.77·17-s + (−0.831 − 0.153i)18-s + (−0.283 + 0.283i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.901 + 0.431i$
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: \((2,\ 400,\ (\ :1/2),\ -0.901 + 0.431i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.399685 - 1.75965i\)
\(L(\frac12)\) \(\approx\) \(0.399685 - 1.75965i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.257 + 1.39i)T \)
5 \( 1 \)
good3 \( 1 + (-1.66 + 1.66i)T - 3iT^{2} \)
7 \( 1 + 2.89iT - 7T^{2} \)
11 \( 1 + (-1.84 - 1.84i)T + 11iT^{2} \)
13 \( 1 + (-3.08 + 3.08i)T - 13iT^{2} \)
17 \( 1 + 7.29T + 17T^{2} \)
19 \( 1 + (1.23 - 1.23i)T - 19iT^{2} \)
23 \( 1 - 4.60iT - 23T^{2} \)
29 \( 1 + (-4.24 + 4.24i)T - 29iT^{2} \)
31 \( 1 - 2.06T + 31T^{2} \)
37 \( 1 + (-1.17 - 1.17i)T + 37iT^{2} \)
41 \( 1 - 4.61iT - 41T^{2} \)
43 \( 1 + (3.03 + 3.03i)T + 43iT^{2} \)
47 \( 1 - 11.7T + 47T^{2} \)
53 \( 1 + (2.73 + 2.73i)T + 53iT^{2} \)
59 \( 1 + (-3.11 - 3.11i)T + 59iT^{2} \)
61 \( 1 + (-2.34 + 2.34i)T - 61iT^{2} \)
67 \( 1 + (8.24 - 8.24i)T - 67iT^{2} \)
71 \( 1 + 3.25iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 0.113T + 79T^{2} \)
83 \( 1 + (9.76 - 9.76i)T - 83iT^{2} \)
89 \( 1 + 3.74iT - 89T^{2} \)
97 \( 1 - 13.9T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.89903836738716142771734857844, −10.10944395855701152951479138355, −8.999935698213366774305200606882, −8.269292737881928155351511940710, −7.30368706637499106915748382769, −6.23103998966698444494103124726, −4.52060682346055134577718877033, −3.59346133785745957259227726989, −2.33398469903124092478264705007, −1.13975530703642885597582911007, 2.65838552906873143485542310802, 3.95063869929337233868078083731, 4.66901387143781327268293418219, 6.05889734994289904893293161289, 6.80113235057587805580161627717, 8.501124379250535943499939557183, 8.785857438899789443123035118334, 9.243268823822094130338782819037, 10.55595484191833151535934466814, 11.68401909897322555597172088581

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