Properties

Label 2-546-91.34-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.856 + 0.515i$
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  + (0.707 + 0.707i)2-s + i·3-s + 1.00i·4-s + (−2.27 + 2.27i)5-s + (−0.707 + 0.707i)6-s + (−2.27 − 1.35i)7-s + (−0.707 + 0.707i)8-s − 9-s − 3.21·10-s + (0.355 − 0.355i)11-s − 1.00·12-s + (−2 − 3i)13-s + (−0.648 − 2.56i)14-s + (−2.27 − 2.27i)15-s − 1.00·16-s + 4.32·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (−1.01 + 1.01i)5-s + (−0.288 + 0.288i)6-s + (−0.858 − 0.512i)7-s + (−0.250 + 0.250i)8-s − 0.333·9-s − 1.01·10-s + (0.107 − 0.107i)11-s − 0.288·12-s + (−0.554 − 0.832i)13-s + (−0.173 − 0.685i)14-s + (−0.586 − 0.586i)15-s − 0.250·16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.856 + 0.515i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.856 + 0.515i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178308 - 0.642338i\)
\(L(\frac12)\) \(\approx\) \(0.178308 - 0.642338i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 - iT \)
7 \( 1 + (2.27 + 1.35i)T \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + (2.27 - 2.27i)T - 5iT^{2} \)
11 \( 1 + (-0.355 + 0.355i)T - 11iT^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + (5.98 - 5.98i)T - 19iT^{2} \)
23 \( 1 + 2.38iT - 23T^{2} \)
29 \( 1 - 1.09T + 29T^{2} \)
31 \( 1 + (1.08 - 1.08i)T - 31iT^{2} \)
37 \( 1 + (5.18 - 5.18i)T - 37iT^{2} \)
41 \( 1 + (3.53 - 3.53i)T - 41iT^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + (-4.71 - 4.71i)T + 47iT^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (3.61 + 3.61i)T + 59iT^{2} \)
61 \( 1 + 4.32iT - 61T^{2} \)
67 \( 1 + (0.531 + 0.531i)T + 67iT^{2} \)
71 \( 1 + (-6.38 - 6.38i)T + 71iT^{2} \)
73 \( 1 + (-5.18 - 5.18i)T + 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (6.71 - 6.71i)T - 83iT^{2} \)
89 \( 1 + (-3.75 - 3.75i)T + 89iT^{2} \)
97 \( 1 + (-12.9 + 12.9i)T - 97iT^{2} \)
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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

−11.22119635801516766509966082828, −10.40592162694200815416269773188, −9.839611261464844290212046206364, −8.317068813277294864814961562888, −7.70514705300580946801869500574, −6.72048394284943791511316786852, −5.94871704967425651551681092484, −4.58071381670826791247292107013, −3.57863053519314764481235666036, −3.04060188597766700051696392052, 0.31261432854847018296460341329, 2.07731796117217031843678152301, 3.44832421261165002470313527916, 4.48424210418940467974105041290, 5.45436000630638271966725043103, 6.62964488132516350036814839764, 7.49513533358234226230251911659, 8.753732384268770195545175088946, 9.188309656977777487441998931651, 10.42511969371927479012082006140

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