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Properties

Label 2-1512-24.11-c1-0-60
Degree $2$
Conductor $1512$
Sign $0.707 - 0.707i$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + 3.12·5-s − i·7-s + (1.99 + 2i)8-s + (4.27 + 1.14i)10-s − 0.397i·11-s + 6.55i·13-s + (0.366 − 1.36i)14-s + (1.99 + 3.46i)16-s + 2i·17-s − 0.527·19-s + (5.42 + 3.12i)20-s + (0.145 − 0.543i)22-s − 8.21·23-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + 1.39·5-s − 0.377i·7-s + (0.707 + 0.707i)8-s + (1.35 + 0.362i)10-s − 0.119i·11-s + 1.81i·13-s + (0.0978 − 0.365i)14-s + (0.499 + 0.866i)16-s + 0.485i·17-s − 0.121·19-s + (1.21 + 0.699i)20-s + (0.0310 − 0.115i)22-s − 1.71·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1512,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.032531429\)
\(L(\frac12)\) \(\approx\) \(4.032531429\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 - 3.12T + 5T^{2} \)
11 \( 1 + 0.397iT - 11T^{2} \)
13 \( 1 - 6.55iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 0.527T + 19T^{2} \)
23 \( 1 + 8.21T + 23T^{2} \)
29 \( 1 - 4.73T + 29T^{2} \)
31 \( 1 + 7.55iT - 31T^{2} \)
37 \( 1 + 3.35iT - 37T^{2} \)
41 \( 1 + 7.48iT - 41T^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 0.795T + 53T^{2} \)
59 \( 1 + 2.47iT - 59T^{2} \)
61 \( 1 + 11.4iT - 61T^{2} \)
67 \( 1 + 6.55T + 67T^{2} \)
71 \( 1 - 2.21T + 71T^{2} \)
73 \( 1 + 5.75T + 73T^{2} \)
79 \( 1 + 10.2iT - 79T^{2} \)
83 \( 1 - 8.36iT - 83T^{2} \)
89 \( 1 - 5.31iT - 89T^{2} \)
97 \( 1 + 19.1T + 97T^{2} \)
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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

−9.639939857495925855788879409724, −8.805994445265903800702807692404, −7.77175965343638852759241655040, −6.82406352359474743617591185828, −6.17464064951042653280183445203, −5.64483342654702894007096871263, −4.45511869561928467640224398072, −3.87247874538648784896907061754, −2.34391312545862750220779274126, −1.79163269784498370686378251787, 1.28181759005472142422045063307, 2.46405066807798917220069160046, 3.07398714975344438113848946420, 4.44081590231075266526536285541, 5.43468243482045049238085457379, 5.80740615599342065898771711702, 6.58372561071678168798834647430, 7.64790540891225122214453415867, 8.599591757361760435516586045798, 9.753898341171027227142075646804

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