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Properties

Label 2-672-56.19-c1-0-10
Degree $2$
Conductor $672$
Sign $-0.746 + 0.665i$
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 + (−0.225 + 0.390i)5-s + (−0.458 − 2.60i)7-s + (0.499 − 0.866i)9-s + (0.360 + 0.623i)11-s − 3.48·13-s − 0.451i·15-s + (−3.55 + 2.05i)17-s + (−3.97 − 2.29i)19-s + (1.69 + 2.02i)21-s + (0.0459 + 0.0265i)23-s + (2.39 + 4.15i)25-s + 0.999i·27-s − 7.85i·29-s + (−4.58 − 7.93i)31-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−0.100 + 0.174i)5-s + (−0.173 − 0.984i)7-s + (0.166 − 0.288i)9-s + (0.108 + 0.188i)11-s − 0.967·13-s − 0.116i·15-s + (−0.862 + 0.498i)17-s + (−0.912 − 0.526i)19-s + (0.370 + 0.442i)21-s + (0.00957 + 0.00552i)23-s + (0.479 + 0.830i)25-s + 0.192i·27-s − 1.45i·29-s + (−0.823 − 1.42i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.746 + 0.665i$
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: \((2,\ 672,\ (\ :1/2),\ -0.746 + 0.665i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.135615 - 0.356059i\)
\(L(\frac12)\) \(\approx\) \(0.135615 - 0.356059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.458 + 2.60i)T \)
good5 \( 1 + (0.225 - 0.390i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.360 - 0.623i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.48T + 13T^{2} \)
17 \( 1 + (3.55 - 2.05i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.97 + 2.29i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.0459 - 0.0265i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 7.85iT - 29T^{2} \)
31 \( 1 + (4.58 + 7.93i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (7.51 + 4.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.94iT - 41T^{2} \)
43 \( 1 + 5.17T + 43T^{2} \)
47 \( 1 + (-0.460 + 0.796i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.71 + 1.56i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.86 + 2.80i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.54 + 4.41i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.93 - 8.54i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 11.1iT - 71T^{2} \)
73 \( 1 + (3.33 - 1.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.40 + 4.85i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.53iT - 83T^{2} \)
89 \( 1 + (12.6 + 7.28i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.14iT - 97T^{2} \)
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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

−10.21811758200352026409316885684, −9.538439730179748164626693011605, −8.467542557033586747716563126645, −7.26530234510081101362900464223, −6.79008685514207509454158739783, −5.61873151318206004169120893398, −4.48873217907883314972642297320, −3.79999982863044861206510251021, −2.18739352542178447241605447533, −0.20273843530244928662100320298, 1.86075337369580232387149766064, 3.08713838937416220554429698272, 4.66689127729654260508133015847, 5.35106689775218957710593806053, 6.47669790676419041417345501635, 7.10703960830592694198608994842, 8.442826411323690884123303046939, 8.929324393475988394139598338449, 10.07156865787320677812102446636, 10.85543486811655687481234088047

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