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Properties

Label 2-20e2-16.13-c1-0-14
Degree $2$
Conductor $400$
Sign $0.988 - 0.150i$
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  + (0.618 − 1.27i)2-s + (2.16 + 2.16i)3-s + (−1.23 − 1.57i)4-s + (4.09 − 1.41i)6-s + 3.30i·7-s + (−2.76 + 0.594i)8-s + 6.40i·9-s + (2.01 − 2.01i)11-s + (0.738 − 6.08i)12-s + (0.794 + 0.794i)13-s + (4.20 + 2.04i)14-s + (−0.955 + 3.88i)16-s + 4.61·17-s + (8.14 + 3.96i)18-s + (−3.48 − 3.48i)19-s + ⋯
L(s)  = 1  + (0.437 − 0.899i)2-s + (1.25 + 1.25i)3-s + (−0.616 − 0.787i)4-s + (1.67 − 0.577i)6-s + 1.24i·7-s + (−0.977 + 0.210i)8-s + 2.13i·9-s + (0.606 − 0.606i)11-s + (0.213 − 1.75i)12-s + (0.220 + 0.220i)13-s + (1.12 + 0.546i)14-s + (−0.238 + 0.971i)16-s + 1.11·17-s + (1.91 + 0.934i)18-s + (−0.800 − 0.800i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $0.988 - 0.150i$
Analytic conductor: \(3.19401\)
Root analytic conductor: \(1.78718\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 400,\ (\ :1/2),\ 0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32553 + 0.176426i\)
\(L(\frac12)\) \(\approx\) \(2.32553 + 0.176426i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.618 + 1.27i)T \)
5 \( 1 \)
good3 \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \)
7 \( 1 - 3.30iT - 7T^{2} \)
11 \( 1 + (-2.01 + 2.01i)T - 11iT^{2} \)
13 \( 1 + (-0.794 - 0.794i)T + 13iT^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + (3.48 + 3.48i)T + 19iT^{2} \)
23 \( 1 + 7.99iT - 23T^{2} \)
29 \( 1 + (1.95 + 1.95i)T + 29iT^{2} \)
31 \( 1 + 5.12T + 31T^{2} \)
37 \( 1 + (-0.448 + 0.448i)T - 37iT^{2} \)
41 \( 1 - 4.02iT - 41T^{2} \)
43 \( 1 + (-4.97 + 4.97i)T - 43iT^{2} \)
47 \( 1 + 5.49T + 47T^{2} \)
53 \( 1 + (3.35 - 3.35i)T - 53iT^{2} \)
59 \( 1 + (-2.07 + 2.07i)T - 59iT^{2} \)
61 \( 1 + (0.557 + 0.557i)T + 61iT^{2} \)
67 \( 1 + (0.636 + 0.636i)T + 67iT^{2} \)
71 \( 1 + 6.85iT - 71T^{2} \)
73 \( 1 + 10.5iT - 73T^{2} \)
79 \( 1 + 17.3T + 79T^{2} \)
83 \( 1 + (-9.48 - 9.48i)T + 83iT^{2} \)
89 \( 1 - 7.62iT - 89T^{2} \)
97 \( 1 - 0.709T + 97T^{2} \)
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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

−11.13049923380822064836619622570, −10.39210601331403996358160532809, −9.349689209784327360043172005609, −8.952344730977179624703490114384, −8.219135892870691865115758169596, −6.16313533229818479075551098089, −5.06547544521573756326770544040, −4.08032327217508791211754541046, −3.11469625764528404665319218582, −2.22740689906662645881754816442, 1.47824700453599774792846152482, 3.33387143943533306356645488013, 4.02918159587653277555652378028, 5.78055972464886954248251957107, 6.92505911191402201517902187369, 7.46130775559178966801392262621, 8.055449650290741883392958466455, 9.094469857015280754989866431899, 9.985582341825470281068818910501, 11.63849355716918562347042659683

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