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Properties

Label 2-672-224.221-c1-0-22
Degree $2$
Conductor $672$
Sign $0.995 + 0.0904i$
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  + (0.534 − 1.30i)2-s + (−0.793 + 0.608i)3-s + (−1.42 − 1.39i)4-s + (−0.913 + 1.19i)5-s + (0.373 + 1.36i)6-s + (2.43 − 1.03i)7-s + (−2.59 + 1.12i)8-s + (0.258 − 0.965i)9-s + (1.07 + 1.83i)10-s + (−0.739 + 5.61i)11-s + (1.98 + 0.239i)12-s + (0.140 − 0.338i)13-s + (−0.0611 − 3.74i)14-s − 1.50i·15-s + (0.0874 + 3.99i)16-s + (1.79 + 1.03i)17-s + ⋯
L(s)  = 1  + (0.377 − 0.925i)2-s + (−0.458 + 0.351i)3-s + (−0.714 − 0.699i)4-s + (−0.408 + 0.532i)5-s + (0.152 + 0.556i)6-s + (0.919 − 0.392i)7-s + (−0.917 + 0.397i)8-s + (0.0862 − 0.321i)9-s + (0.338 + 0.579i)10-s + (−0.223 + 1.69i)11-s + (0.573 + 0.0690i)12-s + (0.0388 − 0.0937i)13-s + (−0.0163 − 0.999i)14-s − 0.387i·15-s + (0.0218 + 0.999i)16-s + (0.435 + 0.251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.995 + 0.0904i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.995 + 0.0904i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39635 - 0.0632983i\)
\(L(\frac12)\) \(\approx\) \(1.39635 - 0.0632983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.534 + 1.30i)T \)
3 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 + (-2.43 + 1.03i)T \)
good5 \( 1 + (0.913 - 1.19i)T + (-1.29 - 4.82i)T^{2} \)
11 \( 1 + (0.739 - 5.61i)T + (-10.6 - 2.84i)T^{2} \)
13 \( 1 + (-0.140 + 0.338i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-1.79 - 1.03i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.95 + 0.652i)T + (18.3 - 4.91i)T^{2} \)
23 \( 1 + (-0.271 + 1.01i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.13 - 1.29i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.47 - 3.22i)T + (-9.57 - 35.7i)T^{2} \)
41 \( 1 + (-2.68 - 2.68i)T + 41iT^{2} \)
43 \( 1 + (-3.95 + 1.63i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (-5.09 + 2.94i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.71 - 12.9i)T + (-51.1 - 13.7i)T^{2} \)
59 \( 1 + (-5.07 - 0.667i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (0.435 + 3.30i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (6.78 - 5.20i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (-4.04 + 4.04i)T - 71iT^{2} \)
73 \( 1 + (12.1 - 3.25i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.03 - 1.75i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.55 + 6.17i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (11.4 + 3.06i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 4.81T + 97T^{2} \)
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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

−10.51276352310682302970788810653, −10.05120784016212336504975550310, −9.092639136645910305102970419453, −7.79164617308889749938766713221, −7.02459571802241057478195748944, −5.60953119771301283817897416121, −4.74732661932958304099546274854, −4.06252560039813374695985267369, −2.79001415311927757972381994907, −1.35019665708549954097082350032, 0.835146319604922136383332310210, 3.03132610412869184116875958713, 4.31346916470553549545058071767, 5.36678547575758935004511910639, 5.76697565580669683577825478887, 6.98434480482582771965205426034, 8.011095444514447241793664084958, 8.368743763978033603652642461769, 9.298928381509746202927014609070, 10.70969296172420901689363850983

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