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Properties

Label 2-20e2-16.11-c2-0-52
Degree $2$
Conductor $400$
Sign $0.382 + 0.924i$
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  + (1.96 + 0.390i)2-s + (−3.81 − 3.81i)3-s + (3.69 + 1.53i)4-s + (−5.99 − 8.97i)6-s + 7.06·7-s + (6.64 + 4.44i)8-s + 20.1i·9-s + (10.3 − 10.3i)11-s + (−8.25 − 19.9i)12-s + (5.72 − 5.72i)13-s + (13.8 + 2.75i)14-s + (11.3 + 11.3i)16-s − 20.9·17-s + (−7.85 + 39.4i)18-s + (−7.58 − 7.58i)19-s + ⋯
L(s)  = 1  + (0.980 + 0.195i)2-s + (−1.27 − 1.27i)3-s + (0.923 + 0.383i)4-s + (−0.999 − 1.49i)6-s + 1.00·7-s + (0.831 + 0.556i)8-s + 2.23i·9-s + (0.942 − 0.942i)11-s + (−0.687 − 1.66i)12-s + (0.440 − 0.440i)13-s + (0.989 + 0.197i)14-s + (0.706 + 0.707i)16-s − 1.23·17-s + (−0.436 + 2.19i)18-s + (−0.398 − 0.398i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.382 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.382 + 0.924i$
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: \((2,\ 400,\ (\ :1),\ 0.382 + 0.924i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.00015 - 1.33750i\)
\(L(\frac12)\) \(\approx\) \(2.00015 - 1.33750i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.390i)T \)
5 \( 1 \)
good3 \( 1 + (3.81 + 3.81i)T + 9iT^{2} \)
7 \( 1 - 7.06T + 49T^{2} \)
11 \( 1 + (-10.3 + 10.3i)T - 121iT^{2} \)
13 \( 1 + (-5.72 + 5.72i)T - 169iT^{2} \)
17 \( 1 + 20.9T + 289T^{2} \)
19 \( 1 + (7.58 + 7.58i)T + 361iT^{2} \)
23 \( 1 - 13.7T + 529T^{2} \)
29 \( 1 + (-32.5 + 32.5i)T - 841iT^{2} \)
31 \( 1 + 26.0iT - 961T^{2} \)
37 \( 1 + (19.6 + 19.6i)T + 1.36e3iT^{2} \)
41 \( 1 + 28.8iT - 1.68e3T^{2} \)
43 \( 1 + (1.30 - 1.30i)T - 1.84e3iT^{2} \)
47 \( 1 - 25.9iT - 2.20e3T^{2} \)
53 \( 1 + (-8.62 - 8.62i)T + 2.80e3iT^{2} \)
59 \( 1 + (43.8 - 43.8i)T - 3.48e3iT^{2} \)
61 \( 1 + (26.3 - 26.3i)T - 3.72e3iT^{2} \)
67 \( 1 + (-62.9 - 62.9i)T + 4.48e3iT^{2} \)
71 \( 1 - 67.9T + 5.04e3T^{2} \)
73 \( 1 + 14.0iT - 5.32e3T^{2} \)
79 \( 1 + 11.0iT - 6.24e3T^{2} \)
83 \( 1 + (-70.2 - 70.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 161. iT - 7.92e3T^{2} \)
97 \( 1 + 125.T + 9.40e3T^{2} \)
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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

−11.16369183611888630456283880149, −10.82069081118456539808744962123, −8.614537906446803199910940989849, −7.76763847712810190909627868756, −6.72062508576777009612635565218, −6.16668746562217109946840643529, −5.25516194534429970834902311553, −4.22894746211628431250724909307, −2.31110162595187438377031428316, −1.01363337669541925209828632546, 1.56800843703158218771112239361, 3.63317635624765397794636353689, 4.75991419124903715296014112639, 4.82009632762800699912660906418, 6.29957004725548522484146587744, 6.87142702774280929624141959759, 8.711733631451339687055311417892, 9.803061854045761239896466197804, 10.73910609896777336538746906194, 11.20520590085859746376608140787

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