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Properties

Label 2-672-168.125-c3-0-35
Degree $2$
Conductor $672$
Sign $-0.402 - 0.915i$
Analytic cond. $39.6492$
Root an. cond. $6.29676$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.70 + 3.63i)3-s + 9.02i·5-s + (−5.01 + 17.8i)7-s + (0.514 − 26.9i)9-s + 8.71·11-s + 52.4·13-s + (−32.8 − 33.4i)15-s + 69.0·17-s + 129.·19-s + (−46.2 − 84.3i)21-s + 177. i·23-s + 43.6·25-s + (96.3 + 101. i)27-s + 247.·29-s − 52.1i·31-s + ⋯
L(s)  = 1  + (−0.713 + 0.700i)3-s + 0.806i·5-s + (−0.271 + 0.962i)7-s + (0.0190 − 0.999i)9-s + 0.238·11-s + 1.11·13-s + (−0.565 − 0.575i)15-s + 0.985·17-s + 1.56·19-s + (−0.480 − 0.876i)21-s + 1.61i·23-s + 0.349·25-s + (0.686 + 0.727i)27-s + 1.58·29-s − 0.302i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.402 - 0.915i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ -0.402 - 0.915i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.779104862\)
\(L(\frac12)\) \(\approx\) \(1.779104862\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.70 - 3.63i)T \)
7 \( 1 + (5.01 - 17.8i)T \)
good5 \( 1 - 9.02iT - 125T^{2} \)
11 \( 1 - 8.71T + 1.33e3T^{2} \)
13 \( 1 - 52.4T + 2.19e3T^{2} \)
17 \( 1 - 69.0T + 4.91e3T^{2} \)
19 \( 1 - 129.T + 6.85e3T^{2} \)
23 \( 1 - 177. iT - 1.21e4T^{2} \)
29 \( 1 - 247.T + 2.43e4T^{2} \)
31 \( 1 + 52.1iT - 2.97e4T^{2} \)
37 \( 1 + 299. iT - 5.06e4T^{2} \)
41 \( 1 + 185.T + 6.89e4T^{2} \)
43 \( 1 - 330. iT - 7.95e4T^{2} \)
47 \( 1 - 81.9T + 1.03e5T^{2} \)
53 \( 1 - 238.T + 1.48e5T^{2} \)
59 \( 1 + 198. iT - 2.05e5T^{2} \)
61 \( 1 + 157.T + 2.26e5T^{2} \)
67 \( 1 + 527. iT - 3.00e5T^{2} \)
71 \( 1 + 245. iT - 3.57e5T^{2} \)
73 \( 1 - 745. iT - 3.89e5T^{2} \)
79 \( 1 + 598.T + 4.93e5T^{2} \)
83 \( 1 + 3.93iT - 5.71e5T^{2} \)
89 \( 1 - 1.25e3T + 7.04e5T^{2} \)
97 \( 1 - 755. iT - 9.12e5T^{2} \)
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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

−10.34921440870781048629759672249, −9.632021863070679579359804143292, −8.916792032124317773010278370770, −7.69288923257403346284022440106, −6.58870116631284495536971306774, −5.83580888828373783489480174122, −5.14523276095338148042864660054, −3.64009823028195202544305226088, −3.02493089240955521064727505682, −1.14312347805165447507383328754, 0.77530948370284885667009198163, 1.23429393096302849114044551060, 3.13434178226142888708395190779, 4.44241252123412921168183442374, 5.29965362528572248752673080946, 6.34795919041194121767998723146, 7.06232523439999513302487156589, 8.070451092156354456667775868409, 8.775352986140914766405863419993, 10.12703380437160624967052912777

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