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Properties

Label 2-672-7.4-c1-0-11
Degree $2$
Conductor $672$
Sign $0.266 + 0.963i$
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  + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s − 0.999·15-s + (4 − 6.92i)17-s + (2 + 3.46i)19-s + (0.500 − 2.59i)21-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s − 0.999·27-s − 5·29-s + (−3.5 + 6.06i)31-s + (0.499 + 0.866i)33-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s − 0.258·15-s + (0.970 − 1.68i)17-s + (0.458 + 0.794i)19-s + (0.109 − 0.566i)21-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s − 0.192·27-s − 0.928·29-s + (−0.628 + 1.08i)31-s + (0.0870 + 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35039 - 1.02732i\)
\(L(\frac12)\) \(\approx\) \(1.35039 - 1.02732i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-4 + 6.92i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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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

−10.35235380394846928489151366286, −9.345334870789155658096372631790, −8.492767421029756162863363909018, −7.60650591330513169942825525092, −7.16758654060059061429937255597, −5.68446161154154989611828484702, −4.86726850298778436145200389917, −3.72124833491832631026741659429, −2.33324262068158988677351796615, −0.964645434429502727272145962107, 1.71915971563612065907288644781, 3.14432682860494934749926411712, 4.08151647151838229731299447282, 5.24967951568593484517241445908, 6.01703606236360138508945314822, 7.50119166433043228449097027836, 7.999251690257549049178788099190, 8.981685868759529814087585806359, 9.785016100286678386750167419489, 10.90552615910957382383820093259

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