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Properties

Label 2-1512-63.25-c1-0-2
Degree $2$
Conductor $1512$
Sign $-0.890 - 0.455i$
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  + (−1.89 + 3.29i)5-s + (−0.841 − 2.50i)7-s + (2.25 + 3.90i)11-s + (0.588 + 1.01i)13-s + (2.95 − 5.12i)17-s + (2.55 + 4.42i)19-s + (−2.09 + 3.62i)23-s + (−4.71 − 8.17i)25-s + (−2.11 + 3.65i)29-s − 6.24·31-s + (9.85 + 1.99i)35-s + (−3.87 − 6.70i)37-s + (−0.754 − 1.30i)41-s + (−5.01 + 8.68i)43-s + 2.23·47-s + ⋯
L(s)  = 1  + (−0.849 + 1.47i)5-s + (−0.318 − 0.948i)7-s + (0.680 + 1.17i)11-s + (0.163 + 0.282i)13-s + (0.717 − 1.24i)17-s + (0.586 + 1.01i)19-s + (−0.435 + 0.755i)23-s + (−0.943 − 1.63i)25-s + (−0.392 + 0.679i)29-s − 1.12·31-s + (1.66 + 0.337i)35-s + (−0.636 − 1.10i)37-s + (−0.117 − 0.204i)41-s + (−0.765 + 1.32i)43-s + 0.326·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.890 - 0.455i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7876485133\)
\(L(\frac12)\) \(\approx\) \(0.7876485133\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.841 + 2.50i)T \)
good5 \( 1 + (1.89 - 3.29i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.25 - 3.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.588 - 1.01i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.95 + 5.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.55 - 4.42i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.09 - 3.62i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.11 - 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 6.24T + 31T^{2} \)
37 \( 1 + (3.87 + 6.70i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.754 + 1.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.01 - 8.68i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + (6.49 - 11.2i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.3T + 59T^{2} \)
61 \( 1 - 1.45T + 61T^{2} \)
67 \( 1 - 1.62T + 67T^{2} \)
71 \( 1 + 8.48T + 71T^{2} \)
73 \( 1 + (-3.72 + 6.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + (-0.307 + 0.532i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.25 - 2.17i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.36 + 4.10i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.827203124345630475819564302274, −9.296604888402395532551068307550, −7.69433237466936761015671766702, −7.42302119463308701762679457282, −6.89285201112999315876961312292, −5.90344002586044482959371095572, −4.57113726223032577370674470206, −3.66309092035006781356828888942, −3.17117806453615290267627173510, −1.62337345848393155910289052184, 0.32054201181512981734059016173, 1.59042726944882814077837136832, 3.23596551554308342638590048354, 3.94093985752826137644588654633, 5.06726806914834831053065627153, 5.69700207058430985357608995086, 6.56940401247884559284150465033, 7.88105905837170738142140084259, 8.473310232270195816292111291141, 8.892804240889199222459062344317

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