Properties

Label 2-672-224.187-c1-0-35
Degree $2$
Conductor $672$
Sign $0.998 + 0.0459i$
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  + (1.36 − 0.369i)2-s + (0.608 − 0.793i)3-s + (1.72 − 1.00i)4-s + (2.34 + 3.05i)5-s + (0.538 − 1.30i)6-s + (−1.86 + 1.87i)7-s + (1.98 − 2.01i)8-s + (−0.258 − 0.965i)9-s + (4.32 + 3.30i)10-s + (0.395 + 3.00i)11-s + (0.252 − 1.98i)12-s + (−0.873 − 2.10i)13-s + (−1.84 + 3.25i)14-s + 3.84·15-s + (1.96 − 3.48i)16-s + (−0.223 − 0.387i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.260i)2-s + (0.351 − 0.458i)3-s + (0.863 − 0.503i)4-s + (1.04 + 1.36i)5-s + (0.219 − 0.533i)6-s + (−0.704 + 0.710i)7-s + (0.702 − 0.711i)8-s + (−0.0862 − 0.321i)9-s + (1.36 + 1.04i)10-s + (0.119 + 0.905i)11-s + (0.0728 − 0.572i)12-s + (−0.242 − 0.584i)13-s + (−0.494 + 0.869i)14-s + 0.993·15-s + (0.492 − 0.870i)16-s + (−0.0543 − 0.0940i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0459i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0459i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.998 + 0.0459i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.998 + 0.0459i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.32063 - 0.0763232i\)
\(L(\frac12)\) \(\approx\) \(3.32063 - 0.0763232i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.369i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (1.86 - 1.87i)T \)
good5 \( 1 + (-2.34 - 3.05i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (-0.395 - 3.00i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (0.873 + 2.10i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (0.223 + 0.387i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.424 + 3.22i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (0.709 - 0.190i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.38 - 5.76i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.58 + 6.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.46 + 4.19i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (7.46 + 7.46i)T + 41iT^{2} \)
43 \( 1 + (8.18 + 3.39i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (2.17 + 1.25i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-13.9 + 1.83i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (0.0581 + 0.441i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.948 - 7.20i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (7.44 + 5.71i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (2.83 - 2.83i)T - 71iT^{2} \)
73 \( 1 + (3.33 - 12.4i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.09 + 3.62i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.63 + 1.92i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (3.30 + 12.3i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 19.4iT - 97T^{2} \)
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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

−10.39932809382876961218807734521, −9.948308333961396083508031764264, −9.006711670066758169530492777526, −7.30435347595142294929860159092, −6.87860031170066599859062823133, −6.01625285237544702363057592522, −5.23713578467849263840321154241, −3.58248083790932401164943294688, −2.65093628131898678759228834706, −2.05844616697044687571149174237, 1.55671759227952127072674224688, 3.05537577241974480232607439039, 4.14414764344945280651439565382, 4.94585250823298490521647149482, 5.90524927928604478955126457622, 6.61357587441190263046564015535, 7.992408882835767356362995736619, 8.738012633509942598859445573550, 9.728505642277942377045761886515, 10.33944516323338428927705521742

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