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Properties

Label 2-20e2-16.11-c2-0-62
Degree $2$
Conductor $400$
Sign $-0.990 + 0.139i$
Analytic cond. $10.8992$
Root an. cond. $3.30139$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 + 1.58i)2-s + (−1.92 − 1.92i)3-s + (−1.05 − 3.85i)4-s + (5.39 − 0.723i)6-s − 4.10·7-s + (7.41 + 3.00i)8-s − 1.58i·9-s + (−0.338 + 0.338i)11-s + (−5.40 + 9.46i)12-s + (11.8 − 11.8i)13-s + (4.98 − 6.53i)14-s + (−13.7 + 8.13i)16-s + 9.17·17-s + (2.51 + 1.92i)18-s + (−5.20 − 5.20i)19-s + ⋯
L(s)  = 1  + (−0.606 + 0.794i)2-s + (−0.641 − 0.641i)3-s + (−0.263 − 0.964i)4-s + (0.899 − 0.120i)6-s − 0.587·7-s + (0.926 + 0.376i)8-s − 0.175i·9-s + (−0.0307 + 0.0307i)11-s + (−0.450 + 0.788i)12-s + (0.912 − 0.912i)13-s + (0.356 − 0.466i)14-s + (−0.861 + 0.508i)16-s + 0.539·17-s + (0.139 + 0.106i)18-s + (−0.273 − 0.273i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.990 + 0.139i$
Analytic conductor: \(10.8992\)
Root analytic conductor: \(3.30139\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1),\ -0.990 + 0.139i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.00838378 - 0.119275i\)
\(L(\frac12)\) \(\approx\) \(0.00838378 - 0.119275i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.21 - 1.58i)T \)
5 \( 1 \)
good3 \( 1 + (1.92 + 1.92i)T + 9iT^{2} \)
7 \( 1 + 4.10T + 49T^{2} \)
11 \( 1 + (0.338 - 0.338i)T - 121iT^{2} \)
13 \( 1 + (-11.8 + 11.8i)T - 169iT^{2} \)
17 \( 1 - 9.17T + 289T^{2} \)
19 \( 1 + (5.20 + 5.20i)T + 361iT^{2} \)
23 \( 1 + 5.92T + 529T^{2} \)
29 \( 1 + (38.8 - 38.8i)T - 841iT^{2} \)
31 \( 1 - 2.05iT - 961T^{2} \)
37 \( 1 + (38.1 + 38.1i)T + 1.36e3iT^{2} \)
41 \( 1 + 40.1iT - 1.68e3T^{2} \)
43 \( 1 + (59.8 - 59.8i)T - 1.84e3iT^{2} \)
47 \( 1 - 57.4iT - 2.20e3T^{2} \)
53 \( 1 + (-16.7 - 16.7i)T + 2.80e3iT^{2} \)
59 \( 1 + (-9.95 + 9.95i)T - 3.48e3iT^{2} \)
61 \( 1 + (-65.8 + 65.8i)T - 3.72e3iT^{2} \)
67 \( 1 + (85.6 + 85.6i)T + 4.48e3iT^{2} \)
71 \( 1 + 70.3T + 5.04e3T^{2} \)
73 \( 1 - 73.9iT - 5.32e3T^{2} \)
79 \( 1 - 114. iT - 6.24e3T^{2} \)
83 \( 1 + (24.0 + 24.0i)T + 6.88e3iT^{2} \)
89 \( 1 + 13.2iT - 7.92e3T^{2} \)
97 \( 1 + 7.27T + 9.40e3T^{2} \)
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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

−10.59890802538259259676256298609, −9.581471099058276213390908219108, −8.711778147606896834440651894771, −7.66200405802972030857663403985, −6.81438063298881258063997214872, −6.01953291689864098978525778640, −5.28444320452299489002695439710, −3.52747428541111898405162083259, −1.40600802226974624964889047640, −0.07291890474415409878800757402, 1.84138567113851834453645334737, 3.47439212197946708381175822787, 4.34795369637475668628370233680, 5.66066929578892714656594425337, 6.84441361943621252969245343004, 8.075138401463375551857608329193, 8.981377555465589693135664204488, 10.01452684113106138435037197501, 10.38107177784361589903095198370, 11.61080890094733148751505855446

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