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Properties

Label 2-1512-63.25-c1-0-8
Degree $2$
Conductor $1512$
Sign $0.694 - 0.719i$
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  + (1.70 − 2.95i)5-s + (−0.410 + 2.61i)7-s + (2.69 + 4.67i)11-s + (1.89 + 3.28i)13-s + (−0.411 + 0.713i)17-s + (0.233 + 0.404i)19-s + (−2.74 + 4.76i)23-s + (−3.30 − 5.72i)25-s + (−0.400 + 0.693i)29-s − 9.90·31-s + (7.01 + 5.66i)35-s + (4.34 + 7.52i)37-s + (−1.84 − 3.19i)41-s + (−4.36 + 7.55i)43-s + 10.4·47-s + ⋯
L(s)  = 1  + (0.761 − 1.31i)5-s + (−0.155 + 0.987i)7-s + (0.813 + 1.40i)11-s + (0.525 + 0.910i)13-s + (−0.0999 + 0.173i)17-s + (0.0535 + 0.0928i)19-s + (−0.573 + 0.993i)23-s + (−0.661 − 1.14i)25-s + (−0.0743 + 0.128i)29-s − 1.77·31-s + (1.18 + 0.957i)35-s + (0.713 + 1.23i)37-s + (−0.288 − 0.498i)41-s + (−0.665 + 1.15i)43-s + 1.53·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.880009613\)
\(L(\frac12)\) \(\approx\) \(1.880009613\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.410 - 2.61i)T \)
good5 \( 1 + (-1.70 + 2.95i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.69 - 4.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.89 - 3.28i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.411 - 0.713i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.233 - 0.404i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.74 - 4.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.400 - 0.693i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 9.90T + 31T^{2} \)
37 \( 1 + (-4.34 - 7.52i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.84 + 3.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.36 - 7.55i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + (-4.71 + 8.17i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.66T + 59T^{2} \)
61 \( 1 - 0.948T + 61T^{2} \)
67 \( 1 - 0.539T + 67T^{2} \)
71 \( 1 - 3.86T + 71T^{2} \)
73 \( 1 + (-2.58 + 4.48i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.82T + 79T^{2} \)
83 \( 1 + (-3.79 + 6.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.73 - 6.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.22 - 5.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.400050312342221984018934016477, −9.033301158908124367455943024023, −8.211647752479547894498150996313, −7.06674519284898094530676507762, −6.18915662990781879300277434088, −5.43194137902669001243736891621, −4.66590767696314157829027134556, −3.75002881562259462727663458073, −2.04316286170593672202233573110, −1.54426355568680822678931696780, 0.76830963652023486448814079659, 2.31982833442247274482587182181, 3.39263547261892518266900824372, 3.93621274866726334176437834394, 5.56597511169778421337541647269, 6.14619674836819913486029387882, 6.86704980872692291011234370976, 7.61164457275341698517984023320, 8.629406895611690841057322700316, 9.448551081632023588556704997194

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