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Properties

Label 2-1512-63.4-c1-0-10
Degree $2$
Conductor $1512$
Sign $0.998 + 0.0616i$
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  − 3.19·5-s + (2.61 − 0.415i)7-s + 2.28·11-s + (−0.675 + 1.16i)13-s + (−2.21 + 3.83i)17-s + (−3.69 − 6.39i)19-s + 6.46·23-s + 5.20·25-s + (1.06 + 1.83i)29-s + (0.316 + 0.547i)31-s + (−8.34 + 1.32i)35-s + (1.92 + 3.34i)37-s + (5.05 − 8.74i)41-s + (4.24 + 7.35i)43-s + (3.26 − 5.65i)47-s + ⋯
L(s)  = 1  − 1.42·5-s + (0.987 − 0.157i)7-s + 0.688·11-s + (−0.187 + 0.324i)13-s + (−0.537 + 0.930i)17-s + (−0.847 − 1.46i)19-s + 1.34·23-s + 1.04·25-s + (0.197 + 0.341i)29-s + (0.0567 + 0.0983i)31-s + (−1.41 + 0.224i)35-s + (0.317 + 0.549i)37-s + (0.788 − 1.36i)41-s + (0.647 + 1.12i)43-s + (0.476 − 0.825i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.998 + 0.0616i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.998 + 0.0616i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.416417288\)
\(L(\frac12)\) \(\approx\) \(1.416417288\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.61 + 0.415i)T \)
good5 \( 1 + 3.19T + 5T^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + (0.675 - 1.16i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.21 - 3.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.69 + 6.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6.46T + 23T^{2} \)
29 \( 1 + (-1.06 - 1.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.316 - 0.547i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.92 - 3.34i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.05 + 8.74i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.24 - 7.35i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.26 + 5.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.39 - 4.15i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.10 - 5.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.45 + 7.71i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.50 - 2.61i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15.3T + 71T^{2} \)
73 \( 1 + (-4.36 + 7.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.938 + 1.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.00 - 5.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.65 + 4.59i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.44 - 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.093354729887662884868311521012, −8.708537537900313396235745051008, −7.86468995316122876184158591631, −7.13326586579402071140554878916, −6.45315081705637977271619946438, −4.99274565253571636880486910127, −4.39420934407347471030837572809, −3.67622771748968401992881093391, −2.32218804167664867139959792257, −0.851666871440073309051528076143, 0.875745798470508686802294104685, 2.37793561189242296414550940782, 3.66207831103907614733880278350, 4.35654879527704613664857045665, 5.14036007836513469513585228530, 6.30982365268531552555078345365, 7.30755601686264618202834459051, 7.86621312911162151866102917956, 8.552310231808609418904540856878, 9.292147159793317848589345619171

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