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Properties

Label 2-672-56.37-c1-0-7
Degree $2$
Conductor $672$
Sign $0.941 - 0.337i$
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 + (3.08 + 1.78i)5-s + (2.38 − 1.14i)7-s + (0.499 − 0.866i)9-s + (3.52 − 2.03i)11-s + 1.44i·13-s − 3.56·15-s + (−3.49 − 6.05i)17-s + (0.261 + 0.150i)19-s + (−1.48 + 2.18i)21-s + (−1.21 + 2.10i)23-s + (3.85 + 6.67i)25-s + 0.999i·27-s − 0.151i·29-s + (−2.37 − 4.11i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (1.38 + 0.797i)5-s + (0.900 − 0.434i)7-s + (0.166 − 0.288i)9-s + (1.06 − 0.613i)11-s + 0.399i·13-s − 0.920·15-s + (−0.847 − 1.46i)17-s + (0.0599 + 0.0345i)19-s + (−0.325 + 0.477i)21-s + (−0.252 + 0.437i)23-s + (0.771 + 1.33i)25-s + 0.192i·27-s − 0.0281i·29-s + (−0.426 − 0.739i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 - 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.941 - 0.337i$
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: \((2,\ 672,\ (\ :1/2),\ 0.941 - 0.337i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81225 + 0.315031i\)
\(L(\frac12)\) \(\approx\) \(1.81225 + 0.315031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.38 + 1.14i)T \)
good5 \( 1 + (-3.08 - 1.78i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.52 + 2.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.44iT - 13T^{2} \)
17 \( 1 + (3.49 + 6.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.261 - 0.150i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.21 - 2.10i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.151iT - 29T^{2} \)
31 \( 1 + (2.37 + 4.11i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-9.82 - 5.67i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.239T + 41T^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + (3.17 - 5.50i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.18 - 2.99i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.73 - 5.62i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.64 + 2.10i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.79 + 2.19i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.46T + 71T^{2} \)
73 \( 1 + (0.284 + 0.493i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.746 - 1.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.0iT - 83T^{2} \)
89 \( 1 + (-1.83 + 3.17i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.4T + 97T^{2} \)
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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

−10.71644218837692807792140084874, −9.504042451485715606791413718843, −9.329034169701527319542058905033, −7.82850922549249853473579517939, −6.72620576122282631249545597183, −6.19003219451649309300464564731, −5.15128596935889245797257019105, −4.16196747461956535890737779949, −2.69928126636657813388452991123, −1.39218339465725944159014034557, 1.43319736046402396419402504425, 2.11624414642092993584393444606, 4.23014817812963059681348620568, 5.10587659851572601818920199525, 5.95723084813539196366214103146, 6.61809203286531667591331852011, 7.982885785837580044546073759308, 8.831733433957874162123413754643, 9.506881303951236101558249732941, 10.49345889704035256576753701993

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