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Properties

Label 2-1512-7.4-c1-0-15
Degree $2$
Conductor $1512$
Sign $0.968 + 0.250i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (−2 − 1.73i)7-s + 3·13-s + (2 − 3.46i)17-s + (−2 − 3.46i)19-s + (1 + 1.73i)23-s + (0.500 − 0.866i)25-s + 4·29-s + (−0.5 + 0.866i)31-s + (0.999 − 5.19i)35-s + (1.5 + 2.59i)37-s + 8·41-s − 43-s + (4 + 6.92i)47-s + (1.00 + 6.92i)49-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (−0.755 − 0.654i)7-s + 0.832·13-s + (0.485 − 0.840i)17-s + (−0.458 − 0.794i)19-s + (0.208 + 0.361i)23-s + (0.100 − 0.173i)25-s + 0.742·29-s + (−0.0898 + 0.155i)31-s + (0.169 − 0.878i)35-s + (0.246 + 0.427i)37-s + 1.24·41-s − 0.152·43-s + (0.583 + 1.01i)47-s + (0.142 + 0.989i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.968 + 0.250i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (865, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.735656867\)
\(L(\frac12)\) \(\approx\) \(1.735656867\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (2 + 1.73i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2 + 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7 + 12.1i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - T + 97T^{2} \)
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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

−9.594853149218903509387141193301, −8.735835669850771880696023393366, −7.71883157969679283570714452914, −6.81155352718207690499549150752, −6.43887373360069500434195140975, −5.41110601630634814384337652317, −4.26971234507418713923348489955, −3.28287098112101589097825422694, −2.50187746479165407039424208507, −0.856072969797229738378922063817, 1.09980481928004140031310309802, 2.32841119679821829379098804471, 3.51550947308607310773471406549, 4.42806140199613045276736698000, 5.78039533657831207745636215121, 5.86113746198257192380044075882, 6.99985720595192438939528840424, 8.160170072091403509105931989072, 8.776081149170347411367135882719, 9.355018318241804348776136873742

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