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Properties

Label 2-672-224.187-c1-0-43
Degree $2$
Conductor $672$
Sign $-0.812 + 0.582i$
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  + (−0.347 − 1.37i)2-s + (0.608 − 0.793i)3-s + (−1.75 + 0.952i)4-s + (−0.430 − 0.561i)5-s + (−1.29 − 0.559i)6-s + (2.64 + 0.0884i)7-s + (1.91 + 2.08i)8-s + (−0.258 − 0.965i)9-s + (−0.620 + 0.785i)10-s + (0.333 + 2.53i)11-s + (−0.315 + 1.97i)12-s + (−2.05 − 4.96i)13-s + (−0.797 − 3.65i)14-s − 0.707·15-s + (2.18 − 3.34i)16-s + (−2.22 − 3.85i)17-s + ⋯
L(s)  = 1  + (−0.245 − 0.969i)2-s + (0.351 − 0.458i)3-s + (−0.879 + 0.476i)4-s + (−0.192 − 0.251i)5-s + (−0.530 − 0.228i)6-s + (0.999 + 0.0334i)7-s + (0.677 + 0.735i)8-s + (−0.0862 − 0.321i)9-s + (−0.196 + 0.248i)10-s + (0.100 + 0.764i)11-s + (−0.0909 + 0.570i)12-s + (−0.570 − 1.37i)13-s + (−0.213 − 0.977i)14-s − 0.182·15-s + (0.546 − 0.837i)16-s + (−0.540 − 0.935i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.812 + 0.582i$
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: \((2,\ 672,\ (\ :1/2),\ -0.812 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.402457 - 1.25146i\)
\(L(\frac12)\) \(\approx\) \(0.402457 - 1.25146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.347 + 1.37i)T \)
3 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 + (-2.64 - 0.0884i)T \)
good5 \( 1 + (0.430 + 0.561i)T + (-1.29 + 4.82i)T^{2} \)
11 \( 1 + (-0.333 - 2.53i)T + (-10.6 + 2.84i)T^{2} \)
13 \( 1 + (2.05 + 4.96i)T + (-9.19 + 9.19i)T^{2} \)
17 \( 1 + (2.22 + 3.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.512 + 3.89i)T + (-18.3 - 4.91i)T^{2} \)
23 \( 1 + (-5.08 + 1.36i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.91 - 7.04i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (3.58 + 6.21i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.42 - 2.62i)T + (9.57 - 35.7i)T^{2} \)
41 \( 1 + (-5.80 - 5.80i)T + 41iT^{2} \)
43 \( 1 + (5.36 + 2.22i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (6.54 + 3.77i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.83 - 0.768i)T + (51.1 - 13.7i)T^{2} \)
59 \( 1 + (-1.01 - 7.73i)T + (-56.9 + 15.2i)T^{2} \)
61 \( 1 + (-1.21 + 9.21i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-5.37 - 4.12i)T + (17.3 + 64.7i)T^{2} \)
71 \( 1 + (-10.2 + 10.2i)T - 71iT^{2} \)
73 \( 1 + (3.11 - 11.6i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-3.74 + 6.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.89 - 1.61i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (1.96 + 7.34i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 - 6.83iT - 97T^{2} \)
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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

−10.22151213972129666366927728202, −9.277794978615640340941565967877, −8.517502873736215866195589183499, −7.76216216261684715981040262429, −6.97612544560469493991341116964, −5.07851341207531319004546326490, −4.68229245066223974943280452469, −3.11639923994904089868544497968, −2.19926400908494821748923673433, −0.78417388761167195223136249562, 1.70469962737465617719073669547, 3.62505115618780562526493600289, 4.53253466655539021803837092704, 5.41449010680916151135976503743, 6.54237755212740125886158230107, 7.42587531598294254210635687037, 8.305749927298388444071979862424, 8.907998636805465107931044605792, 9.739876314093177001484502139642, 10.81052870768023155478284163424

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