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Properties

Label 2-672-168.125-c1-0-2
Degree $2$
Conductor $672$
Sign $-0.925 + 0.377i$
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  + 1.73i·3-s + 3.46i·5-s + (−1 − 2.44i)7-s − 2.99·9-s − 5.65·11-s − 5.99·15-s + (4.24 − 1.73i)21-s − 6.99·25-s − 5.19i·27-s + 2.82·29-s + 4.89i·31-s − 9.79i·33-s + (8.48 − 3.46i)35-s − 10.3i·45-s + (−4.99 + 4.89i)49-s + ⋯
L(s)  = 1  + 0.999i·3-s + 1.54i·5-s + (−0.377 − 0.925i)7-s − 0.999·9-s − 1.70·11-s − 1.54·15-s + (0.925 − 0.377i)21-s − 1.39·25-s − 0.999i·27-s + 0.525·29-s + 0.879i·31-s − 1.70i·33-s + (1.43 − 0.585i)35-s − 1.54i·45-s + (−0.714 + 0.699i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.925 + 0.377i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.925 + 0.377i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108748 - 0.554098i\)
\(L(\frac12)\) \(\approx\) \(0.108748 - 0.554098i\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73iT \)
7 \( 1 + (1 + 2.44i)T \)
good5 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + 5.65T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 - 4.89iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 17.3iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19.5iT - 97T^{2} \)
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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

−10.71887990924501245588517401140, −10.30836294591521628690963184114, −9.682148301465606531396308723199, −8.314173030320880560506098578081, −7.45515519822191585617861638758, −6.60429956932702705885237133666, −5.55878209027610727649949696689, −4.44904382789505592659387527999, −3.29502723690249553425052151119, −2.70062559873554309310896366729, 0.27933571258893938456168005239, 1.88121484176750506394937402220, 2.97395554584487724886649484246, 4.80111988350531236240150447404, 5.47761994803344308810944342912, 6.28163447997644455455584259464, 7.68091415817115243426442094729, 8.228560140457991383541479850426, 8.950688627075467469658273539708, 9.811776287424649449355797282026

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