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Properties

Label 2-672-12.11-c1-0-1
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 − 0.837·13-s + (−2.23 − 1.00i)15-s − 1.64i·17-s + 7.16i·19-s + (−1.58 − 0.707i)21-s − 5.65·23-s + 2.99·25-s + (4.94 − 1.58i)27-s − 7.30i·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 − 0.232·13-s + (−0.577 − 0.258i)15-s − 0.398i·17-s + 1.64i·19-s + (−0.345 − 0.154i)21-s − 1.17·23-s + 0.599·25-s + (0.952 − 0.304i)27-s − 1.35i·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

\(L(1)\) \(\approx\) \(0.0800202 - 0.433753i\)
\(L(\frac12)\) \(\approx\) \(0.0800202 - 0.433753i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.837T + 13T^{2} \)
17 \( 1 + 1.64iT - 17T^{2} \)
19 \( 1 - 7.16iT - 19T^{2} \)
23 \( 1 + 5.65T + 23T^{2} \)
29 \( 1 + 7.30iT - 29T^{2} \)
31 \( 1 + 6.32iT - 31T^{2} \)
37 \( 1 + 8.32T + 37T^{2} \)
41 \( 1 - 1.18iT - 41T^{2} \)
43 \( 1 - 4.32iT - 43T^{2} \)
47 \( 1 + 8.94T + 47T^{2} \)
53 \( 1 - 10.1iT - 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 3.16T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 + 1.18T + 71T^{2} \)
73 \( 1 + 8.32T + 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 - 10.3T + 83T^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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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

−10.78986486339738787059160821050, −10.15020137951685541006769816361, −9.611781333785276167506466341095, −8.340313881283750945940905873016, −7.63246036889221454012008793991, −6.24913645628828298403415881350, −5.62932824361121677814772091516, −4.60358317178487391461030322792, −3.47155215969148118065310607293, −2.40592926154921393596040310944, 0.23209573272376829836668012719, 1.78823754687856873557089327133, 3.09634837561427023532095664143, 4.85901847032516113702176505028, 5.30129247837565988254854679764, 6.62358979215371311400367889524, 7.30449584685710036168309395179, 8.257958441526784122047150481944, 8.896620463884719959563806982208, 10.30535829582622035865303761841

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