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Properties

Label 2-672-168.125-c3-0-14
Degree $2$
Conductor $672$
Sign $0.253 - 0.967i$
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  + (−5.10 − 0.953i)3-s − 17.0i·5-s + (5.04 + 17.8i)7-s + (25.1 + 9.73i)9-s + 37.8·11-s − 77.9·13-s + (−16.2 + 86.8i)15-s + 34.6·17-s − 37.7·19-s + (−8.80 − 95.8i)21-s − 47.8i·23-s − 164.·25-s + (−119. − 73.7i)27-s − 180.·29-s + 163. i·31-s + ⋯
L(s)  = 1  + (−0.983 − 0.183i)3-s − 1.52i·5-s + (0.272 + 0.962i)7-s + (0.932 + 0.360i)9-s + 1.03·11-s − 1.66·13-s + (−0.279 + 1.49i)15-s + 0.494·17-s − 0.456·19-s + (−0.0915 − 0.995i)21-s − 0.434i·23-s − 1.31·25-s + (−0.850 − 0.525i)27-s − 1.15·29-s + 0.949i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.253 - 0.967i$
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: \((2,\ 672,\ (\ :3/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6884909339\)
\(L(\frac12)\) \(\approx\) \(0.6884909339\)
\(L(\frac{5}{2})\) 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 \( 1 + (5.10 + 0.953i)T \)
7 \( 1 + (-5.04 - 17.8i)T \)
good5 \( 1 + 17.0iT - 125T^{2} \)
11 \( 1 - 37.8T + 1.33e3T^{2} \)
13 \( 1 + 77.9T + 2.19e3T^{2} \)
17 \( 1 - 34.6T + 4.91e3T^{2} \)
19 \( 1 + 37.7T + 6.85e3T^{2} \)
23 \( 1 + 47.8iT - 1.21e4T^{2} \)
29 \( 1 + 180.T + 2.43e4T^{2} \)
31 \( 1 - 163. iT - 2.97e4T^{2} \)
37 \( 1 + 159. iT - 5.06e4T^{2} \)
41 \( 1 + 81.5T + 6.89e4T^{2} \)
43 \( 1 - 241. iT - 7.95e4T^{2} \)
47 \( 1 + 356.T + 1.03e5T^{2} \)
53 \( 1 - 585.T + 1.48e5T^{2} \)
59 \( 1 - 172. iT - 2.05e5T^{2} \)
61 \( 1 - 572.T + 2.26e5T^{2} \)
67 \( 1 - 765. iT - 3.00e5T^{2} \)
71 \( 1 - 925. iT - 3.57e5T^{2} \)
73 \( 1 + 590. iT - 3.89e5T^{2} \)
79 \( 1 - 28.4T + 4.93e5T^{2} \)
83 \( 1 - 2.66iT - 5.71e5T^{2} \)
89 \( 1 - 600.T + 7.04e5T^{2} \)
97 \( 1 - 703. iT - 9.12e5T^{2} \)
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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

−10.17400747160899843794036489768, −9.377238351832146466910257567889, −8.690359289235213137665367075309, −7.66674214488301500890836845647, −6.62514710538256688185657643088, −5.49376782044683498066539998579, −5.05210201168692011626916112633, −4.13844424029819518721172877981, −2.13246892000142854432442522874, −1.06005492825406422151301303991, 0.25380028878540771333412792292, 1.89849924966361361724032081668, 3.44166579700632205556453770490, 4.30160704050682198517395843305, 5.44449294817769949598261417048, 6.59130087302489246992725223312, 7.05739241360628887788233185212, 7.76295467931148917340420552794, 9.563382215684882780856121305533, 10.04324603365828938231977763849

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