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Properties

Label 2-672-7.2-c1-0-4
Degree $2$
Conductor $672$
Sign $-0.0854 - 0.996i$
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  + (0.5 + 0.866i)3-s + (−1.37 + 2.37i)5-s + (2.64 − 0.0585i)7-s + (−0.499 + 0.866i)9-s + (0.771 + 1.33i)11-s + 6.03·13-s − 2.74·15-s + (−3.74 − 6.48i)17-s + (−3.01 + 5.22i)19-s + (1.37 + 2.26i)21-s + (−3.74 + 6.48i)23-s + (−1.27 − 2.20i)25-s − 0.999·27-s + 1.25·29-s + (2.64 + 4.58i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.614 + 1.06i)5-s + (0.999 − 0.0221i)7-s + (−0.166 + 0.288i)9-s + (0.232 + 0.403i)11-s + 1.67·13-s − 0.709·15-s + (−0.908 − 1.57i)17-s + (−0.692 + 1.19i)19-s + (0.299 + 0.493i)21-s + (−0.781 + 1.35i)23-s + (−0.254 − 0.440i)25-s − 0.192·27-s + 0.232·29-s + (0.475 + 0.822i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0854 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0854 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.0854 - 0.996i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.0854 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08346 + 1.18032i\)
\(L(\frac12)\) \(\approx\) \(1.08346 + 1.18032i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-2.64 + 0.0585i)T \)
good5 \( 1 + (1.37 - 2.37i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.771 - 1.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.03T + 13T^{2} \)
17 \( 1 + (3.74 + 6.48i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.01 - 5.22i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.74 - 6.48i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (-2.64 - 4.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.47 - 4.28i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.08T + 41T^{2} \)
43 \( 1 + 3.45T + 43T^{2} \)
47 \( 1 + (-4.74 + 8.22i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.91 - 3.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.77 - 4.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.29 + 12.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.01 + 3.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.49T + 71T^{2} \)
73 \( 1 + (6.27 + 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.89 - 6.75i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.52T + 83T^{2} \)
89 \( 1 + (-4.74 + 8.22i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−10.75935421743338275700199460932, −10.06428188958348668479574524136, −8.862160423259817962591067183369, −8.191536240029716002371870012593, −7.29243465275903447864330703045, −6.40131092908350301163045097684, −5.14770163838071184466546251330, −4.04718165133076204803657424869, −3.30740317663983878250352593892, −1.83677340753415590168192409366, 0.893900328245166901811766946596, 2.12498169124565234063860487991, 3.94908938491063240132677140897, 4.48447240508059446606187707422, 5.87065261332527867005514744901, 6.68211860277510966361863437767, 8.067331257812689711019143189707, 8.670178960889794818931316926860, 8.697670393192257870508505044952, 10.52841055033616791617676288983

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