Properties

Label 2-3072-48.29-c0-0-5
Degree $2$
Conductor $3072$
Sign $0.923 - 0.382i$
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  + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)3-s + (1 + i)5-s + 1.41i·7-s − 1.00i·9-s + 1.41·15-s + (1.00 + 1.00i)21-s + i·25-s + (−0.707 − 0.707i)27-s + (1 − i)29-s − 1.41·31-s + (−1.41 + 1.41i)35-s + (1.00 − 1.00i)45-s − 1.00·49-s + (1 + i)53-s + (1.41 + 1.41i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.923 - 0.382i$
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: \((2,\ 3072,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.887843188\)
\(L(\frac12)\) \(\approx\) \(1.887843188\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good5 \( 1 + (-1 - i)T + iT^{2} \)
7 \( 1 - 1.41iT - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1 - i)T + iT^{2} \)
59 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.882882662649836680002026042814, −8.317700262584930625241011855237, −7.32772956605410098486816404949, −6.70674809576638280568441831572, −5.91670545090749496973089760558, −5.53244551160912387720814522941, −4.05031999766707573593851654903, −2.81839659550972779658887060070, −2.54116998413610041546289081331, −1.63983652320964928946055910882, 1.18769800524870273713950342010, 2.17995600944609962171415975583, 3.42409462183955206607210737542, 4.13468988420988663562583351683, 4.96602860971618869624204441395, 5.48624913155813880209990266209, 6.69630270031417476892637764803, 7.41955003435143033852508055462, 8.320892085300359304247127877211, 8.878253786904448552981271102384

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