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Properties

Label 2-1512-24.11-c1-0-46
Degree $2$
Conductor $1512$
Sign $0.243 - 0.969i$
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  + (−0.116 + 1.40i)2-s + (−1.97 − 0.327i)4-s + 2.99·5-s − i·7-s + (0.690 − 2.74i)8-s + (−0.347 + 4.22i)10-s + 5.36i·11-s − 4.55i·13-s + (1.40 + 0.116i)14-s + (3.78 + 1.29i)16-s + 0.958i·17-s + 0.532·19-s + (−5.91 − 0.980i)20-s + (−7.55 − 0.622i)22-s + 2.78·23-s + ⋯
L(s)  = 1  + (−0.0820 + 0.996i)2-s + (−0.986 − 0.163i)4-s + 1.34·5-s − 0.377i·7-s + (0.243 − 0.969i)8-s + (−0.109 + 1.33i)10-s + 1.61i·11-s − 1.26i·13-s + (0.376 + 0.0310i)14-s + (0.946 + 0.322i)16-s + 0.232i·17-s + 0.122·19-s + (−1.32 − 0.219i)20-s + (−1.61 − 0.132i)22-s + 0.581·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.243 - 0.969i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.243 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.912107138\)
\(L(\frac12)\) \(\approx\) \(1.912107138\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.116 - 1.40i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 2.99T + 5T^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + 4.55iT - 13T^{2} \)
17 \( 1 - 0.958iT - 17T^{2} \)
19 \( 1 - 0.532T + 19T^{2} \)
23 \( 1 - 2.78T + 23T^{2} \)
29 \( 1 - 5.15T + 29T^{2} \)
31 \( 1 - 4.66iT - 31T^{2} \)
37 \( 1 - 6.10iT - 37T^{2} \)
41 \( 1 + 12.3iT - 41T^{2} \)
43 \( 1 - 4.45T + 43T^{2} \)
47 \( 1 - 1.40T + 47T^{2} \)
53 \( 1 - 9.57T + 53T^{2} \)
59 \( 1 - 0.0356iT - 59T^{2} \)
61 \( 1 - 5.13iT - 61T^{2} \)
67 \( 1 - 8.59T + 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 6.50T + 73T^{2} \)
79 \( 1 + 1.90iT - 79T^{2} \)
83 \( 1 - 3.13iT - 83T^{2} \)
89 \( 1 - 5.91iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−9.651153554048346232008170417236, −8.833929844842456867549684575203, −7.941366534722412004317565960418, −7.06823242385022995361700886027, −6.53763671674186605004629706913, −5.44389116662307103065674937625, −5.06264049133789081046537857731, −3.91865615367235181281640944978, −2.48999789676423182636423449887, −1.14370497525889033094840618907, 0.990921335674447872032087994135, 2.16122921106291147619054339814, 2.92046580029015985824933302817, 4.06334142636339659403621430827, 5.17010900858147875634467942427, 5.86640785836195781782364841901, 6.65232008299728229925301874326, 8.081885142488006099367006089880, 8.843093019098380529978829326869, 9.407830794792648658791758005513

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