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Properties

Label 2-546-21.20-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.418 + 0.908i$
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  − i·2-s + (−1.35 + 1.07i)3-s − 4-s + 0.745·5-s + (1.07 + 1.35i)6-s + (−2.35 + 1.20i)7-s + i·8-s + (0.697 − 2.91i)9-s − 0.745i·10-s − 2.89i·11-s + (1.35 − 1.07i)12-s + i·13-s + (1.20 + 2.35i)14-s + (−1.01 + 0.799i)15-s + 16-s + 2.53·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.785 + 0.619i)3-s − 0.5·4-s + 0.333·5-s + (0.438 + 0.555i)6-s + (−0.891 + 0.453i)7-s + 0.353i·8-s + (0.232 − 0.972i)9-s − 0.235i·10-s − 0.871i·11-s + (0.392 − 0.309i)12-s + 0.277i·13-s + (0.320 + 0.630i)14-s + (−0.261 + 0.206i)15-s + 0.250·16-s + 0.615·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.418 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.418 + 0.908i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.418 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.383169 - 0.598455i\)
\(L(\frac12)\) \(\approx\) \(0.383169 - 0.598455i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.35 - 1.07i)T \)
7 \( 1 + (2.35 - 1.20i)T \)
13 \( 1 - iT \)
good5 \( 1 - 0.745T + 5T^{2} \)
11 \( 1 + 2.89iT - 11T^{2} \)
17 \( 1 - 2.53T + 17T^{2} \)
19 \( 1 + 4.50iT - 19T^{2} \)
23 \( 1 + 5.62iT - 23T^{2} \)
29 \( 1 + 7.30iT - 29T^{2} \)
31 \( 1 + 6.18iT - 31T^{2} \)
37 \( 1 - 9.19T + 37T^{2} \)
41 \( 1 - 3.79T + 41T^{2} \)
43 \( 1 + 5.86T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + 5.09iT - 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 - 9.20iT - 61T^{2} \)
67 \( 1 - 3.54T + 67T^{2} \)
71 \( 1 - 0.765iT - 71T^{2} \)
73 \( 1 + 6.83iT - 73T^{2} \)
79 \( 1 - 14.3T + 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 + 1.62T + 89T^{2} \)
97 \( 1 - 1.89iT - 97T^{2} \)
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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

−10.54149348302911815467277322152, −9.649596463315919453442376628826, −9.290652270098086934688934651316, −8.063238853630493717660333991240, −6.40979987083053404151386116938, −5.93408032469107112381203638219, −4.76906261724853992554117360009, −3.69346537843244631206472559670, −2.56815496984928208040744835403, −0.47373952575760258581957844043, 1.49412013703907662896748428617, 3.43274216038923850820835233689, 4.83468382893030827872522767693, 5.77287171677051432886076666126, 6.51002184508580456805760699994, 7.37083886307428907065574188619, 8.016054637592763031304942509790, 9.548510201930989682841965611405, 10.01317940574143049492733060796, 11.01167629714928872754078687770

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