Properties

Label 2-672-96.83-c1-0-94
Degree $2$
Conductor $672$
Sign $-0.977 + 0.213i$
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  + (1.21 − 0.724i)2-s + (−0.132 − 1.72i)3-s + (0.949 − 1.76i)4-s + (−0.733 − 1.77i)5-s + (−1.41 − 2.00i)6-s + (−0.707 − 0.707i)7-s + (−0.123 − 2.82i)8-s + (−2.96 + 0.457i)9-s + (−2.17 − 1.61i)10-s + (−0.219 − 0.529i)11-s + (−3.16 − 1.40i)12-s + (5.66 + 2.34i)13-s + (−1.37 − 0.346i)14-s + (−2.96 + 1.50i)15-s + (−2.19 − 3.34i)16-s − 0.0922·17-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (−0.0764 − 0.997i)3-s + (0.474 − 0.880i)4-s + (−0.328 − 0.791i)5-s + (−0.576 − 0.816i)6-s + (−0.267 − 0.267i)7-s + (−0.0437 − 0.999i)8-s + (−0.988 + 0.152i)9-s + (−0.687 − 0.511i)10-s + (−0.0660 − 0.159i)11-s + (−0.913 − 0.405i)12-s + (1.57 + 0.651i)13-s + (−0.366 − 0.0924i)14-s + (−0.764 + 0.387i)15-s + (−0.549 − 0.835i)16-s − 0.0223·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.213i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $-0.977 + 0.213i$
Analytic conductor: \(5.36594\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{672} (659, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 672,\ (\ :1/2),\ -0.977 + 0.213i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.222086 - 2.05964i\)
\(L(\frac12)\) \(\approx\) \(0.222086 - 2.05964i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 + 0.724i)T \)
3 \( 1 + (0.132 + 1.72i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (0.733 + 1.77i)T + (-3.53 + 3.53i)T^{2} \)
11 \( 1 + (0.219 + 0.529i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-5.66 - 2.34i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.0922T + 17T^{2} \)
19 \( 1 + (1.63 - 3.95i)T + (-13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.32 + 1.32i)T + 23iT^{2} \)
29 \( 1 + (2.81 + 1.16i)T + (20.5 + 20.5i)T^{2} \)
31 \( 1 - 0.629iT - 31T^{2} \)
37 \( 1 + (0.995 - 0.412i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-8.34 + 8.34i)T - 41iT^{2} \)
43 \( 1 + (-2.69 + 1.11i)T + (30.4 - 30.4i)T^{2} \)
47 \( 1 - 3.09iT - 47T^{2} \)
53 \( 1 + (-1.53 + 0.637i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-10.1 + 4.20i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (1.88 - 4.53i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (13.7 + 5.71i)T + (47.3 + 47.3i)T^{2} \)
71 \( 1 + (-8.75 + 8.75i)T - 71iT^{2} \)
73 \( 1 + (-5.82 - 5.82i)T + 73iT^{2} \)
79 \( 1 + 9.22T + 79T^{2} \)
83 \( 1 + (15.3 + 6.34i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (0.611 + 0.611i)T + 89iT^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−10.47248899905783612591429066151, −9.123589715979096804993239807865, −8.388177087966119067008124930290, −7.29398805802254106658157586306, −6.26335274938999160193633607038, −5.71791088559839862555336837140, −4.37349638134984859351483592373, −3.50994656496356724335961331719, −2.01273510833217947951831526816, −0.873653698954202361205780868850, 2.74999111027523330134362054708, 3.51319473029710516527748284460, 4.37417972224451666866650335572, 5.56168127760029103927090868941, 6.20780102643051413955080274607, 7.21451112728165209754731230240, 8.297705279745025358514882212548, 9.035186402575782224036077910055, 10.25723153957368990147427894087, 11.22437167638667875059658792449

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