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Properties

Label 2-672-168.5-c1-0-14
Degree $2$
Conductor $672$
Sign $0.966 - 0.257i$
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  + (1.5 − 0.866i)3-s + (3.62 + 2.09i)5-s + (−1.62 + 2.09i)7-s + (1.5 − 2.59i)9-s + (0.0857 + 0.148i)11-s + 7.24·15-s + (−0.621 + 4.54i)21-s + (6.24 + 10.8i)25-s − 5.19i·27-s − 10.4·29-s + (5.37 − 3.10i)31-s + (0.257 + 0.148i)33-s + (−10.2 + 4.18i)35-s + (10.8 − 6.27i)45-s + (−1.74 − 6.77i)49-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (1.61 + 0.935i)5-s + (−0.612 + 0.790i)7-s + (0.5 − 0.866i)9-s + (0.0258 + 0.0448i)11-s + 1.87·15-s + (−0.135 + 0.990i)21-s + (1.24 + 2.16i)25-s − 0.999i·27-s − 1.93·29-s + (0.966 − 0.557i)31-s + (0.0448 + 0.0258i)33-s + (−1.73 + 0.706i)35-s + (1.61 − 0.935i)45-s + (−0.248 − 0.968i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.966 - 0.257i$
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: \((2,\ 672,\ (\ :1/2),\ 0.966 - 0.257i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42437 + 0.318000i\)
\(L(\frac12)\) \(\approx\) \(2.42437 + 0.318000i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (1.62 - 2.09i)T \)
good5 \( 1 + (-3.62 - 2.09i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.0857 - 0.148i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + (-5.37 + 3.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.03 + 8.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.98 + 2.30i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-8.48 + 4.89i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.86 + 6.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.76iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.5iT - 97T^{2} \)
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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

−10.23797436964004370156646254885, −9.522180999888660004197742398596, −9.122649516971267670400884362350, −7.917195669344824873064150527351, −6.81828959383604868659973950526, −6.26428939540097274574169868828, −5.39809686420199043073092378234, −3.53788627396857649604556253293, −2.59089161822093185778108560104, −1.86245757563192936197864714960, 1.42663991338470001005665904708, 2.63361180631381519762635099816, 3.92011922389904386102093492615, 4.93564105989283751496295237148, 5.86031560561407542603762249295, 6.93440410615883246790410576416, 8.070253948662237739836597712718, 9.033390624358757118995293699348, 9.563856715239120112728832619386, 10.13869406372336798557488008088

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