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Properties

Label 2-3072-96.5-c0-0-5
Degree $2$
Conductor $3072$
Sign $0.195 - 0.980i$
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  + (0.382 + 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (−1.30 + 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + 1.84·31-s + (−0.707 − 0.292i)37-s + (1.30 + 1.30i)39-s + (−0.541 + 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)61-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)3-s + (0.541 + 0.541i)7-s + (−0.707 + 0.707i)9-s + (1.70 − 0.707i)13-s + (−1.30 + 0.541i)19-s + (−0.292 + 0.707i)21-s + (0.707 + 0.707i)25-s + (−0.923 − 0.382i)27-s + 1.84·31-s + (−0.707 − 0.292i)37-s + (1.30 + 1.30i)39-s + (−0.541 + 1.30i)43-s − 0.414i·49-s + (−1 − 0.999i)57-s + (−0.707 − 1.70i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.195 - 0.980i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (2945, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.195 - 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.546384809\)
\(L(\frac12)\) \(\approx\) \(1.546384809\)
\(L(1)\) 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 \( 1 + (-0.382 - 0.923i)T \)
good5 \( 1 + (-0.707 - 0.707i)T^{2} \)
7 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (1.30 - 0.541i)T + (0.707 - 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 - 1.84T + T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (0.541 - 1.30i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 - 1.84iT - T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.877577586562685156155730356060, −8.331145441721883964650958857843, −8.053724219209599640982038412125, −6.62721595047507918918665091178, −5.91327057774597595842023209633, −5.15837647806033425390362341002, −4.34745715312598608404957164562, −3.52984534232785803063335439413, −2.72379420524008302338984415808, −1.52827055133606254980378963983, 1.02538880827801870979427363888, 1.92764796764071932154030066015, 2.98714584168916666603824810032, 4.00590364479828792775518949498, 4.70868297867080282857610122283, 6.04711232044242467508324236782, 6.49218553981150172575729028438, 7.17388694497541660879402764629, 8.109539461054794789270015396620, 8.635254787922849039787972318485

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