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Properties

Label 2-1512-63.59-c1-0-23
Degree $2$
Conductor $1512$
Sign $-0.999 - 0.0174i$
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  − 2.04·5-s + (1.41 − 2.23i)7-s − 5.90i·11-s + (−0.139 − 0.0804i)13-s + (−2.77 + 4.81i)17-s + (−4.02 + 2.32i)19-s − 0.433i·23-s − 0.801·25-s + (1.95 − 1.12i)29-s + (−2.57 + 1.48i)31-s + (−2.89 + 4.58i)35-s + (2.17 + 3.77i)37-s + (−2.35 + 4.08i)41-s + (1.82 + 3.15i)43-s + (−0.0650 + 0.112i)47-s + ⋯
L(s)  = 1  − 0.916·5-s + (0.534 − 0.845i)7-s − 1.78i·11-s + (−0.0386 − 0.0223i)13-s + (−0.674 + 1.16i)17-s + (−0.922 + 0.532i)19-s − 0.0903i·23-s − 0.160·25-s + (0.363 − 0.209i)29-s + (−0.463 + 0.267i)31-s + (−0.489 + 0.774i)35-s + (0.358 + 0.620i)37-s + (−0.368 + 0.637i)41-s + (0.278 + 0.481i)43-s + (−0.00948 + 0.0164i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 - 0.0174i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.999 - 0.0174i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3284802075\)
\(L(\frac12)\) \(\approx\) \(0.3284802075\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 2.23i)T \)
good5 \( 1 + 2.04T + 5T^{2} \)
11 \( 1 + 5.90iT - 11T^{2} \)
13 \( 1 + (0.139 + 0.0804i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.77 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.02 - 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 0.433iT - 23T^{2} \)
29 \( 1 + (-1.95 + 1.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.57 - 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 3.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.35 - 4.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.82 - 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.0650 - 0.112i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.7 + 6.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.22 - 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.98 + 3.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.64 + 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.48iT - 71T^{2} \)
73 \( 1 + (2.60 + 1.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.69 - 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.62 + 13.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.04 + 7.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.61 + 1.50i)T + (48.5 - 84.0i)T^{2} \)
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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

−8.738935235223129643335293445716, −8.231929229672704049645036999659, −7.73174568800464056680747831586, −6.56235101280430165127631539464, −5.94156628601303289684415176517, −4.63983677628829958892609218038, −3.94772710799976942225222337220, −3.17060176817036300198857867140, −1.53762066754696465831166836760, −0.12723029940697603657830777279, 1.88991061496325434851803399793, 2.75336175222952428222446871233, 4.26914170274508756509597940146, 4.64418254921831221070336448459, 5.67586824522371305216304433868, 6.93892910586335377622966978614, 7.37436265393521315456177470473, 8.280833208792477359476437504767, 9.074685102995576643260196836786, 9.697414965422173325710770323709

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