Properties

Label 2-3072-48.29-c0-0-9
Degree $2$
Conductor $3072$
Sign $0.382 + 0.923i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
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  + i·3-s − 9-s + (−1 − i)11-s − 2i·17-s + (−1 − i)19-s i·25-s i·27-s + (1 − i)33-s + (−1 + i)43-s + 49-s + 2·51-s + (1 − i)57-s + (1 + i)59-s + (−1 − i)67-s + 75-s + ⋯
L(s)  = 1  + i·3-s − 9-s + (−1 − i)11-s − 2i·17-s + (−1 − i)19-s i·25-s i·27-s + (1 − i)33-s + (−1 + i)43-s + 49-s + 2·51-s + (1 − i)57-s + (1 + i)59-s + (−1 − i)67-s + 75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7191511240\)
\(L(\frac12)\) \(\approx\) \(0.7191511240\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 + iT^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (1 + i)T + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + 2iT - T^{2} \)
19 \( 1 + (1 + i)T + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (1 - i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.748587718119645416211023497258, −8.254672618098637174700978003110, −7.28313698737298633924444932258, −6.37950637268068565718746483443, −5.49873122030689284311615634810, −4.88667237445933256088071658192, −4.17118659043604906272646755827, −2.94330913149798377784932212865, −2.60127571082889161222485459384, −0.41482448358526533172098208651, 1.62655482113198672143716053480, 2.17775377867540877623160208059, 3.40131557006202682967208469783, 4.32599727491055933842671294216, 5.47381964105356767147923368354, 5.98098461228239769989367787400, 6.90524386471279579507214698368, 7.48632183372578108001221370051, 8.334956531924677402425154335216, 8.615127789385884225835831366804

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