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Properties

Label 2-20e2-80.29-c1-0-9
Degree $2$
Conductor $400$
Sign $0.907 - 0.420i$
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  + (−0.889 − 1.09i)2-s + (0.120 − 0.120i)3-s + (−0.418 + 1.95i)4-s + (−0.238 − 0.0252i)6-s + 2.66·7-s + (2.52 − 1.27i)8-s + 2.97i·9-s + (−3.49 + 3.49i)11-s + (0.184 + 0.284i)12-s + (−2.94 + 2.94i)13-s + (−2.37 − 2.93i)14-s + (−3.64 − 1.63i)16-s − 1.85i·17-s + (3.26 − 2.64i)18-s + (3.44 + 3.44i)19-s + ⋯
L(s)  = 1  + (−0.628 − 0.777i)2-s + (0.0692 − 0.0692i)3-s + (−0.209 + 0.977i)4-s + (−0.0974 − 0.0103i)6-s + 1.00·7-s + (0.892 − 0.452i)8-s + 0.990i·9-s + (−1.05 + 1.05i)11-s + (0.0532 + 0.0822i)12-s + (−0.815 + 0.815i)13-s + (−0.634 − 0.784i)14-s + (−0.912 − 0.409i)16-s − 0.448i·17-s + (0.770 − 0.622i)18-s + (0.791 + 0.791i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.907 - 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.907 - 0.420i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.907 - 0.420i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937258 + 0.206890i\)
\(L(\frac12)\) \(\approx\) \(0.937258 + 0.206890i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.889 + 1.09i)T \)
5 \( 1 \)
good3 \( 1 + (-0.120 + 0.120i)T - 3iT^{2} \)
7 \( 1 - 2.66T + 7T^{2} \)
11 \( 1 + (3.49 - 3.49i)T - 11iT^{2} \)
13 \( 1 + (2.94 - 2.94i)T - 13iT^{2} \)
17 \( 1 + 1.85iT - 17T^{2} \)
19 \( 1 + (-3.44 - 3.44i)T + 19iT^{2} \)
23 \( 1 - 0.707T + 23T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 - 6.84T + 31T^{2} \)
37 \( 1 + (-0.0975 - 0.0975i)T + 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-4.43 - 4.43i)T + 43iT^{2} \)
47 \( 1 - 1.89iT - 47T^{2} \)
53 \( 1 + (7.43 + 7.43i)T + 53iT^{2} \)
59 \( 1 + (0.959 - 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (-3.49 + 3.49i)T - 67iT^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 + 15.6T + 73T^{2} \)
79 \( 1 - 6.70T + 79T^{2} \)
83 \( 1 + (-3.87 + 3.87i)T - 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 + 4.79iT - 97T^{2} \)
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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

−11.29133212199544906191436046434, −10.35130863279316395318580447719, −9.776048447307871450177045746165, −8.562424870968853147585150705153, −7.71442151824253795597180412576, −7.20677901739741089866445629168, −5.09379880846379069324733583818, −4.50892005267801030179846967543, −2.68082841700321570356388311170, −1.74694170168276105848129713810, 0.807305768889006223470176554231, 2.83619023657441975020198469293, 4.67123383836066827194681762682, 5.52056493791516540689518106644, 6.52763147318974619568398265663, 7.80860304122643513839388834353, 8.200148080257597198250503319346, 9.270896412508375022006420861782, 10.18299495362128428216886033038, 10.99908860411800337992109034078

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