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Properties

Label 2-1512-21.17-c1-0-17
Degree $2$
Conductor $1512$
Sign $0.165 - 0.986i$
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  + (1.99 + 3.46i)5-s + (0.605 + 2.57i)7-s + (4.23 + 2.44i)11-s + 0.678i·13-s + (3.57 − 6.19i)17-s + (4.19 − 2.42i)19-s + (0.556 − 0.321i)23-s + (−5.49 + 9.52i)25-s − 4.89i·29-s + (4.60 + 2.65i)31-s + (−7.70 + 7.24i)35-s + (−0.0905 − 0.156i)37-s − 9.52·41-s − 2.28·43-s + (1.62 + 2.81i)47-s + ⋯
L(s)  = 1  + (0.894 + 1.54i)5-s + (0.229 + 0.973i)7-s + (1.27 + 0.737i)11-s + 0.188i·13-s + (0.867 − 1.50i)17-s + (0.962 − 0.555i)19-s + (0.116 − 0.0670i)23-s + (−1.09 + 1.90i)25-s − 0.908i·29-s + (0.827 + 0.477i)31-s + (−1.30 + 1.22i)35-s + (−0.0148 − 0.0257i)37-s − 1.48·41-s − 0.348·43-s + (0.237 + 0.410i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323182265\)
\(L(\frac12)\) \(\approx\) \(2.323182265\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.605 - 2.57i)T \)
good5 \( 1 + (-1.99 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.23 - 2.44i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.678iT - 13T^{2} \)
17 \( 1 + (-3.57 + 6.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.19 + 2.42i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.556 + 0.321i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.89iT - 29T^{2} \)
31 \( 1 + (-4.60 - 2.65i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0905 + 0.156i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.52T + 41T^{2} \)
43 \( 1 + 2.28T + 43T^{2} \)
47 \( 1 + (-1.62 - 2.81i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.96 + 3.44i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.94 + 8.57i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.15 - 5.28i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.39 + 9.34i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.03iT - 71T^{2} \)
73 \( 1 + (-0.409 - 0.236i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.08 - 7.07i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.53T + 83T^{2} \)
89 \( 1 + (8.90 + 15.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.07iT - 97T^{2} \)
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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

−9.588067333553366023283870096681, −9.238060499937250790846399848462, −7.936597080635709448876989149979, −6.95050132811326981361080815336, −6.59354120249588859849883838716, −5.62784133255175442225430474287, −4.80419503809866343358086195039, −3.32021829044458824106536033005, −2.65718254880629312543134622228, −1.62751909990298823692179484041, 1.11922646058009259662409036304, 1.49108124269569695313664048530, 3.45503764217551215318687944493, 4.21207331426679290873796321340, 5.22456391485467806373520479121, 5.88795897454157092214382469734, 6.73590492794283692069152251944, 7.994797139218189136702724229135, 8.463940229441262867523709349540, 9.313574991961802970512834750744

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