Properties

Label 2-672-224.221-c1-0-23
Degree $2$
Conductor $672$
Sign $0.687 - 0.726i$
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  + (1.23 − 0.693i)2-s + (−0.793 + 0.608i)3-s + (1.03 − 1.70i)4-s + (−1.06 + 1.38i)5-s + (−0.555 + 1.30i)6-s + (−0.129 + 2.64i)7-s + (0.0962 − 2.82i)8-s + (0.258 − 0.965i)9-s + (−0.350 + 2.44i)10-s + (−0.186 + 1.41i)11-s + (0.215 + 1.98i)12-s + (−1.04 + 2.53i)13-s + (1.67 + 3.34i)14-s − 1.74i·15-s + (−1.84 − 3.55i)16-s + (5.30 + 3.06i)17-s + ⋯
L(s)  = 1  + (0.871 − 0.490i)2-s + (−0.458 + 0.351i)3-s + (0.519 − 0.854i)4-s + (−0.475 + 0.620i)5-s + (−0.226 + 0.530i)6-s + (−0.0490 + 0.998i)7-s + (0.0340 − 0.999i)8-s + (0.0862 − 0.321i)9-s + (−0.110 + 0.773i)10-s + (−0.0561 + 0.426i)11-s + (0.0623 + 0.573i)12-s + (−0.291 + 0.702i)13-s + (0.446 + 0.894i)14-s − 0.451i·15-s + (−0.460 − 0.887i)16-s + (1.28 + 0.742i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.687 - 0.726i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.687 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71839 + 0.739290i\)
\(L(\frac12)\) \(\approx\) \(1.71839 + 0.739290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.23 + 0.693i)T \)
3 \( 1 + (0.793 - 0.608i)T \)
7 \( 1 + (0.129 - 2.64i)T \)
good5 \( 1 + (1.06 - 1.38i)T + (-1.29 - 4.82i)T^{2} \)
11 \( 1 + (0.186 - 1.41i)T + (-10.6 - 2.84i)T^{2} \)
13 \( 1 + (1.04 - 2.53i)T + (-9.19 - 9.19i)T^{2} \)
17 \( 1 + (-5.30 - 3.06i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.44 + 0.848i)T + (18.3 - 4.91i)T^{2} \)
23 \( 1 + (2.16 - 8.07i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (6.71 + 2.78i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 + (2.87 - 4.98i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.86 + 8.94i)T + (-9.57 - 35.7i)T^{2} \)
41 \( 1 + (-0.598 - 0.598i)T + 41iT^{2} \)
43 \( 1 + (1.65 - 0.687i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 + (-1.37 + 0.793i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.46 + 11.1i)T + (-51.1 - 13.7i)T^{2} \)
59 \( 1 + (-2.07 - 0.273i)T + (56.9 + 15.2i)T^{2} \)
61 \( 1 + (-0.482 - 3.66i)T + (-58.9 + 15.7i)T^{2} \)
67 \( 1 + (-2.74 + 2.10i)T + (17.3 - 64.7i)T^{2} \)
71 \( 1 + (5.98 - 5.98i)T - 71iT^{2} \)
73 \( 1 + (-12.4 + 3.34i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-14.3 + 8.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.28 - 3.11i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (13.6 + 3.65i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + 4.06T + 97T^{2} \)
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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

−10.96965465068767311664944117466, −9.694913345427498446465794841700, −9.483437794226354619815259033712, −7.71004092405295443249501693427, −6.96000106397242342901566614181, −5.65368430444196609704531049032, −5.36862090279802662830886063916, −3.91341765435281195154261292576, −3.20593334961140009526386911502, −1.77938697762762546698195418984, 0.838402755987290374808327291371, 2.93680458766906451794319897451, 4.01975942738207522869444567913, 4.99871935085276299349699720499, 5.73853070965126040031982131099, 6.84077341857807079593049344280, 7.77015262829375006375945993804, 8.049265500396400304747040191655, 9.582349371034297135955877488525, 10.63357174867581641255996872363

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