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Properties

Label 2-672-56.37-c1-0-11
Degree $2$
Conductor $672$
Sign $0.811 + 0.584i$
Analytic cond. $5.36594$
Root an. cond. $2.31645$
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  + (0.866 − 0.5i)3-s + (−0.0402 − 0.0232i)5-s + (1.97 − 1.76i)7-s + (0.499 − 0.866i)9-s + (3.11 − 1.79i)11-s + 6.29i·13-s − 0.0464·15-s + (0.258 + 0.447i)17-s + (−2.80 − 1.62i)19-s + (0.823 − 2.51i)21-s + (3.47 − 6.01i)23-s + (−2.49 − 4.32i)25-s − 0.999i·27-s − 2.29i·29-s + (1.05 + 1.82i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.0179 − 0.0103i)5-s + (0.744 − 0.667i)7-s + (0.166 − 0.288i)9-s + (0.939 − 0.542i)11-s + 1.74i·13-s − 0.0119·15-s + (0.0626 + 0.108i)17-s + (−0.644 − 0.371i)19-s + (0.179 − 0.548i)21-s + (0.724 − 1.25i)23-s + (−0.499 − 0.865i)25-s − 0.192i·27-s − 0.426i·29-s + (0.189 + 0.328i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 + 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.811 + 0.584i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.811 + 0.584i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88644 - 0.608840i\)
\(L(\frac12)\) \(\approx\) \(1.88644 - 0.608840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-1.97 + 1.76i)T \)
good5 \( 1 + (0.0402 + 0.0232i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.11 + 1.79i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 6.29iT - 13T^{2} \)
17 \( 1 + (-0.258 - 0.447i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.80 + 1.62i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.47 + 6.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.29iT - 29T^{2} \)
31 \( 1 + (-1.05 - 1.82i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.12 - 0.650i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 - 9.12iT - 43T^{2} \)
47 \( 1 + (-2.32 + 4.02i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.42 - 0.825i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.10 - 0.637i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.72 - 3.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + (-5.57 - 9.64i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.75 + 4.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.25iT - 83T^{2} \)
89 \( 1 + (7.38 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.16T + 97T^{2} \)
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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

−10.49484139109927606160961362422, −9.319964723809573930367572516669, −8.732556324881489661545624729515, −7.87116635653297340835580391840, −6.82835134621646430169054496469, −6.26306802411560840034649248043, −4.51567980044382152892041206416, −4.07464167146056281489634425328, −2.47696218100622431865570071593, −1.22165520719815822601122906274, 1.57570439203705879429685010796, 2.90865010679404514511049326730, 3.99478210680408474013200447517, 5.15716985026761199403301843494, 5.91561275456380040259276800243, 7.36239419872213017594125747379, 7.973534675933430625002240991428, 8.953253789763497870239804429654, 9.563318029210319645663817087629, 10.60447758303796895340524885242

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