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Properties

Label 2-1512-63.25-c1-0-1
Degree $2$
Conductor $1512$
Sign $-0.738 - 0.674i$
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  + (−0.0309 + 0.0536i)5-s + (−0.981 + 2.45i)7-s + (−1.59 − 2.75i)11-s + (−0.252 − 0.437i)13-s + (0.554 − 0.960i)17-s + (0.933 + 1.61i)19-s + (−3.10 + 5.37i)23-s + (2.49 + 4.32i)25-s + (−2.39 + 4.15i)29-s − 2.53·31-s + (−0.101 − 0.128i)35-s + (−4.26 − 7.38i)37-s + (4.94 + 8.56i)41-s + (−3.95 + 6.85i)43-s − 6.58·47-s + ⋯
L(s)  = 1  + (−0.0138 + 0.0240i)5-s + (−0.371 + 0.928i)7-s + (−0.479 − 0.830i)11-s + (−0.0700 − 0.121i)13-s + (0.134 − 0.233i)17-s + (0.214 + 0.370i)19-s + (−0.646 + 1.12i)23-s + (0.499 + 0.865i)25-s + (−0.445 + 0.770i)29-s − 0.455·31-s + (−0.0171 − 0.0217i)35-s + (−0.700 − 1.21i)37-s + (0.772 + 1.33i)41-s + (−0.603 + 1.04i)43-s − 0.960·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.738 - 0.674i$
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: \((2,\ 1512,\ (\ :1/2),\ -0.738 - 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7146407005\)
\(L(\frac12)\) \(\approx\) \(0.7146407005\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.981 - 2.45i)T \)
good5 \( 1 + (0.0309 - 0.0536i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.59 + 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.252 + 0.437i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.554 + 0.960i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.933 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.10 - 5.37i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.39 - 4.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.94 - 8.56i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.95 - 6.85i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 6.58T + 47T^{2} \)
53 \( 1 + (-1.58 + 2.74i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 9.01T + 59T^{2} \)
61 \( 1 + 13.8T + 61T^{2} \)
67 \( 1 - 3.33T + 67T^{2} \)
71 \( 1 + 2.25T + 71T^{2} \)
73 \( 1 + (-2.07 + 3.59i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 2.97T + 79T^{2} \)
83 \( 1 + (2.17 - 3.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.30 - 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.27 - 5.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.523365483730409152838598511804, −9.157576779668207010219988295858, −8.138286146271370214980485061536, −7.51662338723061617239124164291, −6.39116731640742430497065221215, −5.64563782216014228185084598567, −5.03219980014819284229900224311, −3.55763074435250077202308275465, −2.93415147967475418160052704262, −1.60416225750009029442342342611, 0.27048762435933437767151249421, 1.88106570156828555052335782412, 3.05489638759493549723834998818, 4.18007316791994620743093401942, 4.79689195412848720710348478516, 5.99059798766188844803086732489, 6.84115580318703099063475710936, 7.48468568927420636035081727336, 8.310256129905787808306159533589, 9.227903116872460155651725390289

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