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Properties

Label 2-672-56.19-c1-0-11
Degree $2$
Conductor $672$
Sign $0.0596 + 0.998i$
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  + (0.866 − 0.5i)3-s + (−0.128 + 0.222i)5-s + (0.623 − 2.57i)7-s + (0.499 − 0.866i)9-s + (−1.79 − 3.10i)11-s − 4.57·13-s + 0.257i·15-s + (6.92 − 3.99i)17-s + (−0.201 − 0.116i)19-s + (−0.745 − 2.53i)21-s + (−5.76 − 3.32i)23-s + (2.46 + 4.27i)25-s − 0.999i·27-s − 2.80i·29-s + (1.03 + 1.79i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (−0.0575 + 0.0996i)5-s + (0.235 − 0.971i)7-s + (0.166 − 0.288i)9-s + (−0.540 − 0.936i)11-s − 1.27·13-s + 0.0664i·15-s + (1.68 − 0.970i)17-s + (−0.0463 − 0.0267i)19-s + (−0.162 − 0.553i)21-s + (−1.20 − 0.693i)23-s + (0.493 + 0.854i)25-s − 0.192i·27-s − 0.521i·29-s + (0.185 + 0.321i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0596 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0596 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0596 + 0.998i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ 0.0596 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14236 - 1.07617i\)
\(L(\frac12)\) \(\approx\) \(1.14236 - 1.07617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.623 + 2.57i)T \)
good5 \( 1 + (0.128 - 0.222i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.79 + 3.10i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.57T + 13T^{2} \)
17 \( 1 + (-6.92 + 3.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.201 + 0.116i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.76 + 3.32i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.80iT - 29T^{2} \)
31 \( 1 + (-1.03 - 1.79i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.46 - 3.73i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.55iT - 41T^{2} \)
43 \( 1 - 5.42T + 43T^{2} \)
47 \( 1 + (-1.42 + 2.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.93 - 1.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.14 - 1.23i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.44 + 7.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.867 - 1.50i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 8.97iT - 71T^{2} \)
73 \( 1 + (6.57 - 3.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.51 - 4.34i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.79iT - 83T^{2} \)
89 \( 1 + (2.25 + 1.29i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.6iT - 97T^{2} \)
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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

−10.13872007006726893596310437775, −9.624418476429997115210912271765, −8.324319850193503871959783730751, −7.67442814004580798473127426607, −7.06073759698104678470805780918, −5.76908676254319016771687280096, −4.73569078774562607024042659340, −3.52450981590134248324583861611, −2.54511894737657260577067543765, −0.798791497827361172856178907587, 1.93932315050507113892176843791, 2.91296424332467881069510246699, 4.28074995085931122984927098186, 5.20183981664620133350474316756, 6.09174980718964084628350726868, 7.64946812494989095170755805797, 7.906110682762054752604694949931, 9.086960905224689022356441843818, 9.874024349597357486499654549922, 10.38398475581668089113151270466

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