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Properties

Label 2-1512-24.11-c1-0-83
Degree $2$
Conductor $1512$
Sign $0.976 + 0.216i$
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  + (1.41 + 0.102i)2-s + (1.97 + 0.289i)4-s + 2.90·5-s − i·7-s + (2.76 + 0.612i)8-s + (4.10 + 0.298i)10-s − 1.28i·11-s − 1.86i·13-s + (0.102 − 1.41i)14-s + (3.83 + 1.14i)16-s − 3.87i·17-s − 2.04·19-s + (5.75 + 0.843i)20-s + (0.132 − 1.81i)22-s − 0.934·23-s + ⋯
L(s)  = 1  + (0.997 + 0.0726i)2-s + (0.989 + 0.144i)4-s + 1.30·5-s − 0.377i·7-s + (0.976 + 0.216i)8-s + (1.29 + 0.0945i)10-s − 0.388i·11-s − 0.518i·13-s + (0.0274 − 0.376i)14-s + (0.957 + 0.286i)16-s − 0.939i·17-s − 0.468·19-s + (1.28 + 0.188i)20-s + (0.0282 − 0.387i)22-s − 0.194·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 + 0.216i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.976 + 0.216i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.976 + 0.216i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.146013144\)
\(L(\frac12)\) \(\approx\) \(4.146013144\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 - 0.102i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 2.90T + 5T^{2} \)
11 \( 1 + 1.28iT - 11T^{2} \)
13 \( 1 + 1.86iT - 13T^{2} \)
17 \( 1 + 3.87iT - 17T^{2} \)
19 \( 1 + 2.04T + 19T^{2} \)
23 \( 1 + 0.934T + 23T^{2} \)
29 \( 1 - 2.98T + 29T^{2} \)
31 \( 1 - 1.85iT - 31T^{2} \)
37 \( 1 - 3.85iT - 37T^{2} \)
41 \( 1 - 8.72iT - 41T^{2} \)
43 \( 1 + 1.19T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 - 10.9iT - 59T^{2} \)
61 \( 1 + 4.70iT - 61T^{2} \)
67 \( 1 - 4.41T + 67T^{2} \)
71 \( 1 + 9.72T + 71T^{2} \)
73 \( 1 - 9.40T + 73T^{2} \)
79 \( 1 - 16.4iT - 79T^{2} \)
83 \( 1 - 3.72iT - 83T^{2} \)
89 \( 1 + 12.2iT - 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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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

−9.745131481865633899080532869107, −8.557451461425110129107523568903, −7.68365392065998352324902863286, −6.65304031637364295688112685566, −6.16498410534882770542603192933, −5.26367435639368046230447881323, −4.61165324555725721771784670200, −3.32332403130030812657070453487, −2.52413078228744288270805811456, −1.35217561011953689251481664799, 1.73803954566804012938615119442, 2.26365632494811957328710779142, 3.52619339855586921935604774380, 4.56951287057940369077187519647, 5.39469944847083005809107183180, 6.18744107990628588582360374270, 6.63306240215105032386658431557, 7.75663349877967601979922762225, 8.785800290452097484047048384737, 9.692249648081764621535937992625

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