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Properties

Label 2-546-91.5-c1-0-6
Degree $2$
Conductor $546$
Sign $0.720 + 0.693i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 + 0.499i)4-s + (3.26 − 0.876i)5-s + (0.707 + 0.707i)6-s + (−1.99 + 1.73i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−1.69 − 2.93i)10-s + (1.42 − 5.32i)11-s + (0.5 − 0.866i)12-s + (0.661 + 3.54i)13-s + (2.19 + 1.48i)14-s + (−2.39 + 2.39i)15-s + (0.500 − 0.866i)16-s + (1.21 + 2.10i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (1.46 − 0.391i)5-s + (0.288 + 0.288i)6-s + (−0.755 + 0.655i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.535 − 0.927i)10-s + (0.429 − 1.60i)11-s + (0.144 − 0.249i)12-s + (0.183 + 0.983i)13-s + (0.585 + 0.396i)14-s + (−0.618 + 0.618i)15-s + (0.125 − 0.216i)16-s + (0.295 + 0.511i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.720 + 0.693i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24827 - 0.503153i\)
\(L(\frac12)\) \(\approx\) \(1.24827 - 0.503153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (1.99 - 1.73i)T \)
13 \( 1 + (-0.661 - 3.54i)T \)
good5 \( 1 + (-3.26 + 0.876i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.42 + 5.32i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.21 - 2.10i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.32 + 0.889i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.824 - 0.476i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.96T + 29T^{2} \)
31 \( 1 + (-0.160 + 0.599i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-2.34 + 0.628i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (7.08 + 7.08i)T + 41iT^{2} \)
43 \( 1 + 8.62iT - 43T^{2} \)
47 \( 1 + (0.140 + 0.523i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.63 - 4.56i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-8.99 - 2.41i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-12.1 - 7.01i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 + 3.15i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.83 - 8.83i)T - 71iT^{2} \)
73 \( 1 + (2.76 + 0.741i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-0.573 + 0.993i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (2.46 + 2.46i)T + 83iT^{2} \)
89 \( 1 + (1.18 + 4.42i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-4.97 - 4.97i)T + 97iT^{2} \)
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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

−10.48253773921030317847128964487, −9.949136346772358261995216871718, −8.905034359339152949399515816617, −8.768255012168787610571768310201, −6.72028716115148518223595725831, −5.91640196881190707101702363720, −5.28061918526257010509505636195, −3.78799288181046683908001082922, −2.59790328090895187918766359399, −1.14175581130723011700571243293, 1.27674690787067536804822807339, 2.90820973292836994928068141439, 4.64397126089093896255775250792, 5.57209024971448806155538960424, 6.56203363432450687059693471056, 6.93295429738366497122201643264, 8.015335817225696895276368303370, 9.502883615506964678496953512007, 9.962258765752480771671685623831, 10.42055175280124231741789560474

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