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Properties

Label 2-20e2-80.29-c1-0-3
Degree $2$
Conductor $400$
Sign $-0.998 - 0.0589i$
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  + (−0.320 + 1.37i)2-s + (0.720 − 0.720i)3-s + (−1.79 − 0.883i)4-s + (0.761 + 1.22i)6-s − 4.02·7-s + (1.79 − 2.18i)8-s + 1.96i·9-s + (−0.646 + 0.646i)11-s + (−1.92 + 0.656i)12-s + (−4.91 + 4.91i)13-s + (1.29 − 5.54i)14-s + (2.43 + 3.17i)16-s + 2.70i·17-s + (−2.70 − 0.629i)18-s + (0.438 + 0.438i)19-s + ⋯
L(s)  = 1  + (−0.226 + 0.973i)2-s + (0.416 − 0.416i)3-s + (−0.897 − 0.441i)4-s + (0.310 + 0.499i)6-s − 1.52·7-s + (0.633 − 0.773i)8-s + 0.653i·9-s + (−0.195 + 0.195i)11-s + (−0.557 + 0.189i)12-s + (−1.36 + 1.36i)13-s + (0.345 − 1.48i)14-s + (0.609 + 0.792i)16-s + 0.656i·17-s + (−0.636 − 0.148i)18-s + (0.100 + 0.100i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.998 - 0.0589i$
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: \((2,\ 400,\ (\ :1/2),\ -0.998 - 0.0589i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0150957 + 0.511921i\)
\(L(\frac12)\) \(\approx\) \(0.0150957 + 0.511921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.320 - 1.37i)T \)
5 \( 1 \)
good3 \( 1 + (-0.720 + 0.720i)T - 3iT^{2} \)
7 \( 1 + 4.02T + 7T^{2} \)
11 \( 1 + (0.646 - 0.646i)T - 11iT^{2} \)
13 \( 1 + (4.91 - 4.91i)T - 13iT^{2} \)
17 \( 1 - 2.70iT - 17T^{2} \)
19 \( 1 + (-0.438 - 0.438i)T + 19iT^{2} \)
23 \( 1 + 3.60T + 23T^{2} \)
29 \( 1 + (2.00 + 2.00i)T + 29iT^{2} \)
31 \( 1 - 4.30T + 31T^{2} \)
37 \( 1 + (-0.743 - 0.743i)T + 37iT^{2} \)
41 \( 1 + 0.603iT - 41T^{2} \)
43 \( 1 + (5.03 + 5.03i)T + 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-4.07 - 4.07i)T + 53iT^{2} \)
59 \( 1 + (1.22 - 1.22i)T - 59iT^{2} \)
61 \( 1 + (6.98 + 6.98i)T + 61iT^{2} \)
67 \( 1 + (-5.24 + 5.24i)T - 67iT^{2} \)
71 \( 1 - 13.7iT - 71T^{2} \)
73 \( 1 - 1.30T + 73T^{2} \)
79 \( 1 + 0.611T + 79T^{2} \)
83 \( 1 + (1.29 - 1.29i)T - 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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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

−11.98796010687765722369513891147, −10.31766344519330557135813539152, −9.761144323636673958969189164151, −8.900974305206264715866999605387, −7.86595867370595483376783199127, −7.02171794016956317637360539180, −6.37160315353290884021192210947, −5.10798656212996676265992709738, −3.88710862857170588378679506407, −2.22490441718348766112016034945, 0.32605910401827250324721605998, 2.81448967958429500597683404787, 3.25238090155508771628230460449, 4.58240142944248731152398376697, 5.89438012143247437297336444323, 7.27198728567040561482747089618, 8.345554286760079962991324838564, 9.532135129687231210161402403873, 9.742507501744688653854661419264, 10.54137061199971950624757355593

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