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Properties

Label 2-1620-15.14-c2-0-13
Degree $2$
Conductor $1620$
Sign $-0.278 - 0.960i$
Analytic cond. $44.1418$
Root an. cond. $6.64392$
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  + (−4.80 + 1.39i)5-s + 12.3i·7-s − 4.62i·11-s − 1.26i·13-s + 22.9·17-s + 32.8·19-s + 0.797·23-s + (21.1 − 13.3i)25-s − 39.6i·29-s + 1.46·31-s + (−17.2 − 59.4i)35-s + 44.9i·37-s + 39.1i·41-s + 36.3i·43-s + 61.0·47-s + ⋯
L(s)  = 1  + (−0.960 + 0.278i)5-s + 1.76i·7-s − 0.420i·11-s − 0.0971i·13-s + 1.34·17-s + 1.73·19-s + 0.0346·23-s + (0.844 − 0.534i)25-s − 1.36i·29-s + 0.0471·31-s + (−0.492 − 1.69i)35-s + 1.21i·37-s + 0.954i·41-s + 0.846i·43-s + 1.29·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
Sign: $-0.278 - 0.960i$
Analytic conductor: \(44.1418\)
Root analytic conductor: \(6.64392\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1620} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1620,\ (\ :1),\ -0.278 - 0.960i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.535463013\)
\(L(\frac12)\) \(\approx\) \(1.535463013\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (4.80 - 1.39i)T \)
good7 \( 1 - 12.3iT - 49T^{2} \)
11 \( 1 + 4.62iT - 121T^{2} \)
13 \( 1 + 1.26iT - 169T^{2} \)
17 \( 1 - 22.9T + 289T^{2} \)
19 \( 1 - 32.8T + 361T^{2} \)
23 \( 1 - 0.797T + 529T^{2} \)
29 \( 1 + 39.6iT - 841T^{2} \)
31 \( 1 - 1.46T + 961T^{2} \)
37 \( 1 - 44.9iT - 1.36e3T^{2} \)
41 \( 1 - 39.1iT - 1.68e3T^{2} \)
43 \( 1 - 36.3iT - 1.84e3T^{2} \)
47 \( 1 - 61.0T + 2.20e3T^{2} \)
53 \( 1 + 13.9T + 2.80e3T^{2} \)
59 \( 1 - 58.0iT - 3.48e3T^{2} \)
61 \( 1 + 55.0T + 3.72e3T^{2} \)
67 \( 1 - 66.9iT - 4.48e3T^{2} \)
71 \( 1 - 70.6iT - 5.04e3T^{2} \)
73 \( 1 - 23.1iT - 5.32e3T^{2} \)
79 \( 1 + 29.1T + 6.24e3T^{2} \)
83 \( 1 + 130.T + 6.88e3T^{2} \)
89 \( 1 + 48.0iT - 7.92e3T^{2} \)
97 \( 1 - 19.3iT - 9.40e3T^{2} \)
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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

−9.460087340845937019393774021233, −8.476673864858106023870330417573, −7.995697848362464508072734821892, −7.18957562080760302773979446460, −5.97317949754081594186191197393, −5.53393696217062530794608356684, −4.47047987755116088317513289929, −3.17809237978284322984956921858, −2.78075784958908189596068245153, −1.12663787707818440094352691697, 0.51964260646965960530737279464, 1.36601909579143001084031837267, 3.29767072346966801979990488776, 3.76823559236440835852241789872, 4.72156286496503206303122153194, 5.52025684769387846530935367096, 6.99348540285743115477453676375, 7.40533724183352375801332843614, 7.85947157782261004992899758519, 9.012783304415675952245177705187

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