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Properties

Label 2-1404-9.4-c1-0-3
Degree $2$
Conductor $1404$
Sign $0.925 - 0.377i$
Analytic cond. $11.2109$
Root an. cond. $3.34828$
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  + (−0.348 + 0.603i)5-s + (−1.09 − 1.90i)7-s + (2.04 + 3.54i)11-s + (0.5 − 0.866i)13-s − 1.99·17-s + 2.80·19-s + (1.46 − 2.54i)23-s + (2.25 + 3.90i)25-s + (0.380 + 0.658i)29-s + (−1.51 + 2.62i)31-s + 1.53·35-s + 3.64·37-s + (3.73 − 6.46i)41-s + (4.13 + 7.16i)43-s + (2.77 + 4.80i)47-s + ⋯
L(s)  = 1  + (−0.155 + 0.270i)5-s + (−0.415 − 0.719i)7-s + (0.617 + 1.06i)11-s + (0.138 − 0.240i)13-s − 0.484·17-s + 0.644·19-s + (0.306 − 0.530i)23-s + (0.451 + 0.781i)25-s + (0.0706 + 0.122i)29-s + (−0.272 + 0.471i)31-s + 0.258·35-s + 0.599·37-s + (0.583 − 1.01i)41-s + (0.630 + 1.09i)43-s + (0.404 + 0.701i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1404 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1404 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1404\)    =    \(2^{2} \cdot 3^{3} \cdot 13\)
Sign: $0.925 - 0.377i$
Analytic conductor: \(11.2109\)
Root analytic conductor: \(3.34828\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1404} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1404,\ (\ :1/2),\ 0.925 - 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.580910467\)
\(L(\frac12)\) \(\approx\) \(1.580910467\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good5 \( 1 + (0.348 - 0.603i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.09 + 1.90i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.04 - 3.54i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.99T + 17T^{2} \)
19 \( 1 - 2.80T + 19T^{2} \)
23 \( 1 + (-1.46 + 2.54i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.380 - 0.658i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.51 - 2.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 3.64T + 37T^{2} \)
41 \( 1 + (-3.73 + 6.46i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.13 - 7.16i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.77 - 4.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.15T + 53T^{2} \)
59 \( 1 + (-2.16 + 3.75i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.68 - 6.38i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.10 + 1.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.60T + 71T^{2} \)
73 \( 1 - 7.07T + 73T^{2} \)
79 \( 1 + (2.66 + 4.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.91 - 15.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.14T + 89T^{2} \)
97 \( 1 + (-9.11 - 15.7i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.550837600143911863464776787765, −8.999681289068919384387316592024, −7.82122861115616527051274298549, −7.09192776747809381350994659025, −6.60069273292357940694919817060, −5.41794876756634481042921858602, −4.39440625411494120441535948110, −3.64288168967219244422412899635, −2.50257196263718973925879612004, −1.05298818575888486421624965651, 0.832515431854962717580496077497, 2.38193126396203621629112623140, 3.41370939685024266613243175678, 4.34670980305104270600988589660, 5.50822731013070826733159775919, 6.12725651134116325316811559339, 7.00394040017909730639783237249, 8.035723516325097513067024746260, 8.887659848808307822675574976323, 9.237439307408480198724863208456

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