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Properties

Label 2-672-7.5-c2-0-18
Degree $2$
Conductor $672$
Sign $0.977 + 0.211i$
Analytic cond. $18.3106$
Root an. cond. $4.27909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (6.36 + 3.67i)5-s + (−6.34 + 2.96i)7-s + (1.5 − 2.59i)9-s + (−5.90 − 10.2i)11-s − 22.8i·13-s − 12.7·15-s + (18.3 − 10.5i)17-s + (18.3 + 10.5i)19-s + (6.94 − 9.93i)21-s + (17.1 − 29.6i)23-s + (14.4 + 25.0i)25-s + 5.19i·27-s + 8.51·29-s + (−9.70 + 5.60i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (1.27 + 0.734i)5-s + (−0.905 + 0.423i)7-s + (0.166 − 0.288i)9-s + (−0.537 − 0.930i)11-s − 1.75i·13-s − 0.848·15-s + (1.07 − 0.621i)17-s + (0.964 + 0.556i)19-s + (0.330 − 0.473i)21-s + (0.744 − 1.28i)23-s + (0.578 + 1.00i)25-s + 0.192i·27-s + 0.293·29-s + (−0.312 + 0.180i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.977 + 0.211i$
Analytic conductor: \(18.3106\)
Root analytic conductor: \(4.27909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1),\ 0.977 + 0.211i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.706089859\)
\(L(\frac12)\) \(\approx\) \(1.706089859\)
\(L(2)\) 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 \( 1 + (1.5 - 0.866i)T \)
7 \( 1 + (6.34 - 2.96i)T \)
good5 \( 1 + (-6.36 - 3.67i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (5.90 + 10.2i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 22.8iT - 169T^{2} \)
17 \( 1 + (-18.3 + 10.5i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (-18.3 - 10.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-17.1 + 29.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 8.51T + 841T^{2} \)
31 \( 1 + (9.70 - 5.60i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (21.6 - 37.5i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 10.4iT - 1.68e3T^{2} \)
43 \( 1 + 2.53T + 1.84e3T^{2} \)
47 \( 1 + (-60.2 - 34.7i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-17.4 - 30.1i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-45.7 + 26.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-22.1 - 12.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-16.3 - 28.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 107.T + 5.04e3T^{2} \)
73 \( 1 + (-74.7 + 43.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (-66.7 + 115. i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 26.5iT - 6.88e3T^{2} \)
89 \( 1 + (46.0 + 26.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + 168. iT - 9.40e3T^{2} \)
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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

−10.27542762887441549485339182845, −9.752405097591264889800720478914, −8.696314785104844129113194594551, −7.53354499068352390195602172773, −6.42535911644141995785960050777, −5.69431758862819692733323483159, −5.27755705384119565854924516785, −3.17936773778631058439266738190, −2.84138795664518542994759576905, −0.77196361466814635008695407540, 1.16066480057997665261698765461, 2.20118838132061180812321864673, 3.85071204256351124325612751975, 5.12489060239179985087740526021, 5.71003623075234290692857945503, 6.83990382970980341702821001301, 7.36778442132363933123348366274, 8.915220911397681927489890027613, 9.624167568632621929563561874727, 10.03467165414559450064800777017

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