Properties

Degree $2$
Conductor $672$
Sign $0.895 - 0.444i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (1.60 − 2.77i)5-s + (1.02 + 2.43i)7-s + (−0.499 + 0.866i)9-s + (2.12 + 3.68i)11-s − 3.15·13-s + 3.20·15-s + (2.20 + 3.81i)17-s + (1.57 − 2.73i)19-s + (−1.60 + 2.10i)21-s + (2.20 − 3.81i)23-s + (−2.62 − 4.54i)25-s − 0.999·27-s + 7.20·29-s + (1.02 + 1.77i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.715 − 1.23i)5-s + (0.387 + 0.922i)7-s + (−0.166 + 0.288i)9-s + (0.640 + 1.10i)11-s − 0.874·13-s + 0.826·15-s + (0.533 + 0.924i)17-s + (0.361 − 0.626i)19-s + (−0.349 + 0.459i)21-s + (0.459 − 0.795i)23-s + (−0.524 − 0.909i)25-s − 0.192·27-s + 1.33·29-s + (0.183 + 0.318i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.895 - 0.444i$
Motivic weight: \(1\)
Character: $\chi_{672} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91470 + 0.449103i\)
\(L(\frac12)\) \(\approx\) \(1.91470 + 0.449103i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.02 - 2.43i)T \)
good5 \( 1 + (-1.60 + 2.77i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.12 - 3.68i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.15T + 13T^{2} \)
17 \( 1 + (-2.20 - 3.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.57 + 2.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.20 + 3.81i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.20T + 29T^{2} \)
31 \( 1 + (-1.02 - 1.77i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.82 + 8.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 + 0.750T + 43T^{2} \)
47 \( 1 + (1.20 - 2.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.64 - 2.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.12 - 7.14i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.04 + 7.01i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.57 - 4.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6.40T + 71T^{2} \)
73 \( 1 + (7.62 + 13.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.22 - 14.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 14.5T + 83T^{2} \)
89 \( 1 + (1.20 - 2.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−10.21026052491103808927680447473, −9.651186765605900428424178892654, −8.866461487606094211454112789611, −8.353336911022718713273748261125, −7.06403389836155659576986209857, −5.81754230807467243115559357529, −4.95740265844783163784361505607, −4.39920482762315674455241326155, −2.65514470273238601080802599721, −1.54843430853644033002120335281, 1.23441540075090375389614989861, 2.74160861253324596500372140267, 3.51559420616996438489286187651, 5.05590272631229646629392714726, 6.22577817912384605192802324243, 6.93414111874194099809089870239, 7.63007939682604076101330019051, 8.619129259484074885472082361517, 9.957805959925595175384192768831, 10.14358384325962659148755018752

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