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Properties

Label 2-672-168.149-c0-0-1
Degree $2$
Conductor $672$
Sign $0.832 - 0.553i$
Analytic cond. $0.335371$
Root an. cond. $0.579112$
Motivic weight $0$
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 + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 0.999·15-s + (−0.499 + 0.866i)21-s − 0.999·27-s − 29-s + (−0.5 − 0.866i)31-s + (0.499 − 0.866i)33-s + 0.999·35-s + (0.499 + 0.866i)45-s + (−0.499 + 0.866i)49-s + (0.5 + 0.866i)53-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + 0.999·15-s + (−0.499 + 0.866i)21-s − 0.999·27-s − 29-s + (−0.5 − 0.866i)31-s + (0.499 − 0.866i)33-s + 0.999·35-s + (0.499 + 0.866i)45-s + (−0.499 + 0.866i)49-s + (0.5 + 0.866i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.335371\)
Root analytic conductor: \(0.579112\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (401, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.171609900\)
\(L(\frac12)\) \(\approx\) \(1.171609900\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−10.81062538326327691669575886295, −9.691033409159790931732633059033, −9.063708239301009610618427229504, −8.469444121854547520638316484297, −7.65080337952761155978342878634, −5.85593767120683256892465347312, −5.37966868091882362727087198527, −4.43936632309463661566148667803, −3.15307568079583808449612955107, −1.96179931205958697876066291763, 1.67152980012138698261184570844, 2.71114177024210431121502880609, 3.92006594587286479499080429671, 5.28550417542179721972182787911, 6.49294543615988995286632276312, 7.19760192463088888829712604857, 7.75622842397281138326330756989, 8.825158967966344825043179155360, 9.887364876054883765997833013985, 10.56960542980599139432910063008

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