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Properties

Label 2-672-12.11-c1-0-5
Degree $2$
Conductor $672$
Sign $0.577 - 0.816i$
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  + (−1.70 + 0.292i)3-s − 0.585i·5-s + i·7-s + (2.82 − i)9-s − 2·11-s + 0.585·13-s + (0.171 + i)15-s + 0.828i·17-s + 2.24i·19-s + (−0.292 − 1.70i)21-s + 4·23-s + 4.65·25-s + (−4.53 + 2.53i)27-s + 0.828i·29-s + 6.82i·31-s + ⋯
L(s)  = 1  + (−0.985 + 0.169i)3-s − 0.261i·5-s + 0.377i·7-s + (0.942 − 0.333i)9-s − 0.603·11-s + 0.162·13-s + (0.0442 + 0.258i)15-s + 0.200i·17-s + 0.514i·19-s + (−0.0639 − 0.372i)21-s + 0.834·23-s + 0.931·25-s + (−0.872 + 0.487i)27-s + 0.153i·29-s + 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.868955 + 0.449804i\)
\(L(\frac12)\) \(\approx\) \(0.868955 + 0.449804i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.292i)T \)
7 \( 1 - iT \)
good5 \( 1 + 0.585iT - 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 - 0.585T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 - 2.24iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 - 6.82iT - 31T^{2} \)
37 \( 1 - 4.82T + 37T^{2} \)
41 \( 1 - 10iT - 41T^{2} \)
43 \( 1 - 6.48iT - 43T^{2} \)
47 \( 1 - 9.65T + 47T^{2} \)
53 \( 1 - 9.31iT - 53T^{2} \)
59 \( 1 + 2.24T + 59T^{2} \)
61 \( 1 - 5.75T + 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 8.82T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 8.58T + 83T^{2} \)
89 \( 1 - 3.65iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−10.77750424007027063692157148993, −9.893240463615416832062953250410, −9.010339597518731778552149353916, −8.013811188695822000298190554651, −6.96071656128207588959829549560, −6.05555522058257940805077250026, −5.20407485283440960723994260811, −4.41497978454695720328962575441, −2.98018689520208026215155644446, −1.22787886560167106695665827260, 0.69783016129076199385879112074, 2.44043697319980317257542342853, 3.95668556758420235371392159104, 5.01339835540800440140581583751, 5.82802787239449586360302673728, 6.92847772272591650269064830278, 7.41843034126456560991944332068, 8.633876686451767927168759637462, 9.740565499707030256295599113658, 10.58401355709675055852680340937

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