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Properties

Label 2-672-21.20-c3-0-20
Degree $2$
Conductor $672$
Sign $0.334 - 0.942i$
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  + (3.46 − 3.87i)3-s + 1.11·5-s + (−7.01 − 17.1i)7-s + (−3.00 − 26.8i)9-s + 43.6i·11-s + 87.0i·13-s + (3.85 − 4.30i)15-s − 60.8·17-s + 118. i·19-s + (−90.6 − 32.1i)21-s − 52.3i·23-s − 123.·25-s + (−114. − 81.3i)27-s + 208. i·29-s + 51.9i·31-s + ⋯
L(s)  = 1  + (0.666 − 0.745i)3-s + 0.0995·5-s + (−0.378 − 0.925i)7-s + (−0.111 − 0.993i)9-s + 1.19i·11-s + 1.85i·13-s + (0.0663 − 0.0741i)15-s − 0.868·17-s + 1.42i·19-s + (−0.942 − 0.334i)21-s − 0.474i·23-s − 0.990·25-s + (−0.814 − 0.579i)27-s + 1.33i·29-s + 0.300i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.334 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.334 - 0.942i$
Analytic conductor: \(39.6492\)
Root analytic conductor: \(6.29676\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :3/2),\ 0.334 - 0.942i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.458463624\)
\(L(\frac12)\) \(\approx\) \(1.458463624\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.46 + 3.87i)T \)
7 \( 1 + (7.01 + 17.1i)T \)
good5 \( 1 - 1.11T + 125T^{2} \)
11 \( 1 - 43.6iT - 1.33e3T^{2} \)
13 \( 1 - 87.0iT - 2.19e3T^{2} \)
17 \( 1 + 60.8T + 4.91e3T^{2} \)
19 \( 1 - 118. iT - 6.85e3T^{2} \)
23 \( 1 + 52.3iT - 1.21e4T^{2} \)
29 \( 1 - 208. iT - 2.43e4T^{2} \)
31 \( 1 - 51.9iT - 2.97e4T^{2} \)
37 \( 1 - 171.T + 5.06e4T^{2} \)
41 \( 1 + 51.9T + 6.89e4T^{2} \)
43 \( 1 - 351.T + 7.95e4T^{2} \)
47 \( 1 - 354.T + 1.03e5T^{2} \)
53 \( 1 + 578. iT - 1.48e5T^{2} \)
59 \( 1 + 336.T + 2.05e5T^{2} \)
61 \( 1 - 255. iT - 2.26e5T^{2} \)
67 \( 1 + 26.9T + 3.00e5T^{2} \)
71 \( 1 - 1.15e3iT - 3.57e5T^{2} \)
73 \( 1 + 763. iT - 3.89e5T^{2} \)
79 \( 1 - 470.T + 4.93e5T^{2} \)
83 \( 1 + 1.05e3T + 5.71e5T^{2} \)
89 \( 1 + 1.00e3T + 7.04e5T^{2} \)
97 \( 1 + 820. iT - 9.12e5T^{2} \)
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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

−10.03921888793028715304128095949, −9.415352025381132161794831648750, −8.558605453078425890434751820841, −7.41280774940533853562832239041, −6.96101791831520028921418016241, −6.14489061285944376290276828237, −4.42883641560200093374140288060, −3.79632962775729110489742254191, −2.25526110264931954429490069527, −1.42998136891114126626186307712, 0.36054260955709009935219526148, 2.49876063120836453340386085495, 3.04934957107886311057803592351, 4.26426529918391044361921300773, 5.50910047841934782470740156609, 6.02197540016347718007094227275, 7.61033343482080820346648709663, 8.353079807230446064442682325150, 9.115501250243291790988285296776, 9.750132775524362685173257743772

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