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Properties

Label 2-672-12.11-c1-0-23
Degree $2$
Conductor $672$
Sign $-0.934 + 0.356i$
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  + (0.707 − 1.58i)3-s − 1.41i·5-s + i·7-s + (−2.00 − 2.23i)9-s − 4.47·11-s − 7.16·13-s + (−2.23 − 1.00i)15-s − 7.30i·17-s + 0.837i·19-s + (1.58 + 0.707i)21-s + 5.65·23-s + 2.99·25-s + (−4.94 + 1.58i)27-s − 1.64i·29-s + 6.32i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.912i)3-s − 0.632i·5-s + 0.377i·7-s + (−0.666 − 0.745i)9-s − 1.34·11-s − 1.98·13-s + (−0.577 − 0.258i)15-s − 1.77i·17-s + 0.192i·19-s + (0.345 + 0.154i)21-s + 1.17·23-s + 0.599·25-s + (−0.952 + 0.304i)27-s − 0.305i·29-s + 1.13i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.934 + 0.356i$
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: \((2,\ 672,\ (\ :1/2),\ -0.934 + 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178320 - 0.966597i\)
\(L(\frac12)\) \(\approx\) \(0.178320 - 0.966597i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 1.58i)T \)
7 \( 1 - iT \)
good5 \( 1 + 1.41iT - 5T^{2} \)
11 \( 1 + 4.47T + 11T^{2} \)
13 \( 1 + 7.16T + 13T^{2} \)
17 \( 1 + 7.30iT - 17T^{2} \)
19 \( 1 - 0.837iT - 19T^{2} \)
23 \( 1 - 5.65T + 23T^{2} \)
29 \( 1 + 1.64iT - 29T^{2} \)
31 \( 1 - 6.32iT - 31T^{2} \)
37 \( 1 - 4.32T + 37T^{2} \)
41 \( 1 + 10.1iT - 41T^{2} \)
43 \( 1 + 8.32iT - 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 + 1.18iT - 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 3.16T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 4.32T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 7.53T + 83T^{2} \)
89 \( 1 + 1.18iT - 89T^{2} \)
97 \( 1 + 10.6T + 97T^{2} \)
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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

−9.897303144573287517927530047090, −9.179305283288325857641616406767, −8.341012768643968675024422656487, −7.39591315237539197024622410086, −6.94599380835986193236998139339, −5.26740138267717700547329139967, −5.00624298894617178978774223207, −2.97557803021906792295090887207, −2.29858261603519240425404158228, −0.45951354822095963822983817039, 2.41562304396615535181400716163, 3.17653944485991332451710318583, 4.52151204689432941265831724335, 5.17575738597635070062296597413, 6.46939106857623855430117744011, 7.64955527945946435504999753575, 8.126245974680700130480785585077, 9.441170933425563387469955594813, 10.05457660438176651028779921522, 10.71830857802738436694821934283

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