Properties

Label 2-12e3-48.29-c0-0-1
Degree $2$
Conductor $1728$
Sign $-0.382 + 0.923i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)11-s + (−1 − i)13-s − 1.41i·17-s − 31-s + (0.707 − 0.707i)35-s + (1 − i)43-s − 1.41i·47-s + (0.707 + 0.707i)53-s + 1.00i·55-s + 1.41i·65-s + (−1 − i)67-s − 1.41·71-s + i·73-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)5-s + i·7-s + (−0.707 − 0.707i)11-s + (−1 − i)13-s − 1.41i·17-s − 31-s + (0.707 − 0.707i)35-s + (1 − i)43-s − 1.41i·47-s + (0.707 + 0.707i)53-s + 1.00i·55-s + 1.41i·65-s + (−1 − i)67-s − 1.41·71-s + i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6225010762\)
\(L(\frac12)\) \(\approx\) \(0.6225010762\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
89 \( 1 + 1.41T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.057867352122886034524152962680, −8.560579965738067299988881220403, −7.72696061553846992310968482973, −7.12857042998466860450758679788, −5.61786269038940756551980183896, −5.37494337854309505315929996393, −4.39978722555397598192815622328, −3.13588063935268142939853697068, −2.38905442266244460134343487945, −0.45533861842714877033138907550, 1.76854226414883281379709208327, 2.96373279465201844086410963084, 4.06423382406103284418400831694, 4.52027588007608767883237291715, 5.79240394399931854780880881073, 6.86525923532066924452135072959, 7.38570991363460634499789284727, 7.87231198254763514618610439783, 9.015104862174181682584268285524, 9.910429611275954150473681940093

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