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Properties

Label 2-672-168.5-c1-0-24
Degree $2$
Conductor $672$
Sign $-0.961 + 0.273i$
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  + (−0.390 − 1.68i)3-s + (1.54 + 0.894i)5-s + (−2.63 − 0.230i)7-s + (−2.69 + 1.31i)9-s + (−0.501 − 0.868i)11-s − 2.47·13-s + (0.904 − 2.96i)15-s + (−3.32 − 5.76i)17-s + (1.85 − 3.22i)19-s + (0.641 + 4.53i)21-s + (−6.85 − 3.95i)23-s + (−0.900 − 1.55i)25-s + (3.27 + 4.03i)27-s + 0.748·29-s + (−2.87 + 1.65i)31-s + ⋯
L(s)  = 1  + (−0.225 − 0.974i)3-s + (0.692 + 0.399i)5-s + (−0.996 − 0.0869i)7-s + (−0.898 + 0.439i)9-s + (−0.151 − 0.261i)11-s − 0.685·13-s + (0.233 − 0.765i)15-s + (−0.807 − 1.39i)17-s + (0.426 − 0.738i)19-s + (0.139 + 0.990i)21-s + (−1.42 − 0.824i)23-s + (−0.180 − 0.311i)25-s + (0.630 + 0.776i)27-s + 0.138·29-s + (−0.515 + 0.297i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.961 + 0.273i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0882405 - 0.632862i\)
\(L(\frac12)\) \(\approx\) \(0.0882405 - 0.632862i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.390 + 1.68i)T \)
7 \( 1 + (2.63 + 0.230i)T \)
good5 \( 1 + (-1.54 - 0.894i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.501 + 0.868i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.47T + 13T^{2} \)
17 \( 1 + (3.32 + 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.85 + 3.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.85 + 3.95i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.748T + 29T^{2} \)
31 \( 1 + (2.87 - 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.22 - 1.86i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.01T + 41T^{2} \)
43 \( 1 - 9.19iT - 43T^{2} \)
47 \( 1 + (1.19 - 2.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.33 + 10.9i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.34 + 4.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.02 + 3.50i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.89 - 3.98i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.46iT - 71T^{2} \)
73 \( 1 + (5.68 - 3.28i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.53 - 4.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.65iT - 83T^{2} \)
89 \( 1 + (7.39 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.75iT - 97T^{2} \)
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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

−9.987652461778234289165643496121, −9.414039502491089794331654485809, −8.274303447617980189614982707769, −7.21588376637944132812581444462, −6.58810353374315714270750022206, −5.88894149186487743440338236787, −4.73674869506152282318774748482, −2.97084767097954768636089154328, −2.26408622321145395629134353529, −0.31881077968978785945884451212, 2.08766025947324374730926437486, 3.52218515018123469629803894200, 4.38244463791618323725326562051, 5.73735617118488503613097754766, 5.96800071033309066465849490816, 7.38039013905329473697948207385, 8.606179193450600933427133088581, 9.415381496178092866235253862706, 9.978498943153193732857823467980, 10.55944187865729257934139288320

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