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Properties

Label 2-672-56.19-c1-0-13
Degree $2$
Conductor $672$
Sign $0.0638 + 0.997i$
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 + (1.61 − 2.79i)5-s + (−1.82 − 1.91i)7-s + (0.499 − 0.866i)9-s + (1.10 + 1.91i)11-s + 5.08·13-s − 3.22i·15-s + (−2.73 + 1.57i)17-s + (−2.93 − 1.69i)19-s + (−2.53 − 0.743i)21-s + (−2.65 − 1.53i)23-s + (−2.70 − 4.69i)25-s − 0.999i·27-s − 9.88i·29-s + (1.01 + 1.75i)31-s + ⋯
L(s)  = 1  + (0.499 − 0.288i)3-s + (0.721 − 1.25i)5-s + (−0.690 − 0.723i)7-s + (0.166 − 0.288i)9-s + (0.333 + 0.577i)11-s + 1.40·13-s − 0.833i·15-s + (−0.663 + 0.383i)17-s + (−0.673 − 0.388i)19-s + (−0.554 − 0.162i)21-s + (−0.553 − 0.319i)23-s + (−0.541 − 0.938i)25-s − 0.192i·27-s − 1.83i·29-s + (0.182 + 0.315i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0638 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0638 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(672\)    =    \(2^{5} \cdot 3 \cdot 7\)
Sign: $0.0638 + 0.997i$
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: \((2,\ 672,\ (\ :1/2),\ 0.0638 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36205 - 1.27764i\)
\(L(\frac12)\) \(\approx\) \(1.36205 - 1.27764i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.82 + 1.91i)T \)
good5 \( 1 + (-1.61 + 2.79i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.10 - 1.91i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.08T + 13T^{2} \)
17 \( 1 + (2.73 - 1.57i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.93 + 1.69i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.65 + 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.88iT - 29T^{2} \)
31 \( 1 + (-1.01 - 1.75i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.798 + 0.460i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.96iT - 41T^{2} \)
43 \( 1 - 6.68T + 43T^{2} \)
47 \( 1 + (1.06 - 1.83i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.12 - 1.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-10.6 + 6.14i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.34 - 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.40 - 7.63i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.7iT - 71T^{2} \)
73 \( 1 + (-7.82 + 4.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.60 - 5.54i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.57iT - 83T^{2} \)
89 \( 1 + (6.32 + 3.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 15.2iT - 97T^{2} \)
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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

−10.07594355054260957365116158620, −9.339977377136437396499423999982, −8.675708441449583180806885017476, −7.86502205040004656077272121393, −6.55228836437397327591937323468, −6.04770769376043730106223130345, −4.55446054114757420353615936967, −3.85365478627166377486861513322, −2.20973577675256316594294482250, −0.995733397619025911809698202723, 2.01875976711825331329085039220, 3.07873524296110395642632978956, 3.82264304419658413021463247157, 5.54884164035556428404032881550, 6.31298204975233665502207080516, 6.93930840294689082681796984278, 8.372023903403442056928612326288, 9.015986607257815792241795531547, 9.823153973753315537385095890067, 10.73582125514379427646189643351

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