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Properties

Label 2-672-168.5-c1-0-5
Degree $2$
Conductor $672$
Sign $0.0972 - 0.995i$
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  + (1.52 + 0.815i)3-s + (0.461 + 0.266i)5-s + (−0.489 + 2.60i)7-s + (1.66 + 2.49i)9-s + (2.28 + 3.96i)11-s − 4.97·13-s + (0.487 + 0.783i)15-s + (−2.16 − 3.74i)17-s + (−0.921 + 1.59i)19-s + (−2.86 + 3.57i)21-s + (−0.103 − 0.0596i)23-s + (−2.35 − 4.08i)25-s + (0.516 + 5.17i)27-s + 7.74·29-s + (1.93 − 1.11i)31-s + ⋯
L(s)  = 1  + (0.882 + 0.470i)3-s + (0.206 + 0.119i)5-s + (−0.184 + 0.982i)7-s + (0.556 + 0.830i)9-s + (0.689 + 1.19i)11-s − 1.38·13-s + (0.125 + 0.202i)15-s + (−0.524 − 0.907i)17-s + (−0.211 + 0.366i)19-s + (−0.625 + 0.779i)21-s + (−0.0215 − 0.0124i)23-s + (−0.471 − 0.816i)25-s + (0.0993 + 0.995i)27-s + 1.43·29-s + (0.347 − 0.200i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0972 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0972 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0972 - 0.995i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.0972 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43538 + 1.30201i\)
\(L(\frac12)\) \(\approx\) \(1.43538 + 1.30201i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.815i)T \)
7 \( 1 + (0.489 - 2.60i)T \)
good5 \( 1 + (-0.461 - 0.266i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.28 - 3.96i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.97T + 13T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.921 - 1.59i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.103 + 0.0596i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.74T + 29T^{2} \)
31 \( 1 + (-1.93 + 1.11i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.02 - 4.05i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 - 1.87iT - 43T^{2} \)
47 \( 1 + (-2.91 + 5.04i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.29 - 3.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.71 + 3.29i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 + 1.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.4 - 6.05i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.20iT - 71T^{2} \)
73 \( 1 + (-8.35 + 4.82i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0228 + 0.0396i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.86iT - 83T^{2} \)
89 \( 1 + (-8.23 + 14.2i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7.18iT - 97T^{2} \)
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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

−10.30460990200825146957426905750, −9.715910465888680667754510588476, −9.190260777470554247668942724106, −8.214512559978423192088059897803, −7.28476269578183419104961971780, −6.35769706531062154714197762638, −4.95363374191117956679885971328, −4.33329154103045279318789457436, −2.76682008921634213426741605226, −2.15821568903259837618009565998, 0.967662582039063644664739198858, 2.45925865962241025795489805601, 3.61094981281538524748658867337, 4.51133235815778763096372542994, 6.06529978684783450586782209013, 6.87048469870333009829787302207, 7.69583271022885853131462619688, 8.564096270773115905697666296041, 9.363998024976228841921439690065, 10.15476233085672485224825044442

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