Properties

Label 2-546-13.9-c1-0-3
Degree $2$
Conductor $546$
Sign $-0.597 - 0.802i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
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  + (−0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.561·5-s + (−0.499 − 0.866i)6-s + (0.5 + 0.866i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.280 + 0.486i)10-s + (0.780 − 1.35i)11-s + 0.999·12-s + (−0.5 + 3.57i)13-s − 0.999·14-s + (−0.280 + 0.486i)15-s + (−0.5 + 0.866i)16-s + (2.5 + 4.33i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.251·5-s + (−0.204 − 0.353i)6-s + (0.188 + 0.327i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.0887 + 0.153i)10-s + (0.235 − 0.407i)11-s + 0.288·12-s + (−0.138 + 0.990i)13-s − 0.267·14-s + (−0.0724 + 0.125i)15-s + (−0.125 + 0.216i)16-s + (0.606 + 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.597 - 0.802i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.597 - 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.457330 + 0.910517i\)
\(L(\frac12)\) \(\approx\) \(0.457330 + 0.910517i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 3.57i)T \)
good5 \( 1 - 0.561T + 5T^{2} \)
11 \( 1 + (-0.780 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.34 - 4.05i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.438 + 0.759i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 + (5.40 - 9.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.06 + 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.56 - 9.63i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 4.68T + 47T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + (2.43 + 4.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.56 + 6.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2 - 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 - 6.24T + 83T^{2} \)
89 \( 1 + (1.34 - 2.32i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.12 - 3.67i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.98431376127137458061113300189, −10.02079375663931826564285898518, −9.395415676237713516870414259646, −8.478917209501164380039184759865, −7.62946073692979952448935302284, −6.33848361340608420257078706634, −5.79539041791953508252543694692, −4.68374896571471523754025836180, −3.55009733553347817523035492660, −1.64597133344588705047313029374, 0.72578109763899831388137551844, 2.19526854342959724842261703529, 3.43418833114470502007861530368, 4.86851453958344796612352960406, 5.77507233372255184933654097995, 7.25937245960365133829550218255, 7.59052919865805356299634279700, 8.907442645800777824251033631063, 9.683292928449028746029524910911, 10.56644008207606664160290429562

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