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Properties

Label 2-546-21.5-c1-0-23
Degree $2$
Conductor $546$
Sign $0.846 - 0.532i$
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  + (0.866 + 0.5i)2-s + (1.08 + 1.35i)3-s + (0.499 + 0.866i)4-s + (1.49 − 2.59i)5-s + (0.259 + 1.71i)6-s + (−0.0366 − 2.64i)7-s + 0.999i·8-s + (−0.663 + 2.92i)9-s + (2.59 − 1.49i)10-s + (1.10 − 0.638i)11-s + (−0.631 + 1.61i)12-s + i·13-s + (1.29 − 2.30i)14-s + (5.12 − 0.776i)15-s + (−0.5 + 0.866i)16-s + (0.544 + 0.942i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.624 + 0.781i)3-s + (0.249 + 0.433i)4-s + (0.669 − 1.16i)5-s + (0.105 + 0.699i)6-s + (−0.0138 − 0.999i)7-s + 0.353i·8-s + (−0.221 + 0.975i)9-s + (0.820 − 0.473i)10-s + (0.333 − 0.192i)11-s + (−0.182 + 0.465i)12-s + 0.277i·13-s + (0.345 − 0.617i)14-s + (1.32 − 0.200i)15-s + (−0.125 + 0.216i)16-s + (0.132 + 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.846 - 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.846 - 0.532i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.846 - 0.532i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62811 + 0.757141i\)
\(L(\frac12)\) \(\approx\) \(2.62811 + 0.757141i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.08 - 1.35i)T \)
7 \( 1 + (0.0366 + 2.64i)T \)
13 \( 1 - iT \)
good5 \( 1 + (-1.49 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.10 + 0.638i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.544 - 0.942i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6.52 - 3.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (5.47 + 3.15i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.01iT - 29T^{2} \)
31 \( 1 + (7.20 - 4.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.363 + 0.629i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.09T + 41T^{2} \)
43 \( 1 - 8.73T + 43T^{2} \)
47 \( 1 + (-0.962 + 1.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.80 - 5.08i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.18 + 2.04i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.29 + 3.05i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.68 - 9.85i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.18iT - 71T^{2} \)
73 \( 1 + (9.69 - 5.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.66 + 11.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.63T + 83T^{2} \)
89 \( 1 + (-6.56 + 11.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.3iT - 97T^{2} \)
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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

−10.71857462656833568789786129839, −9.866544627551991827999946303616, −9.162223543269702081145495862775, −8.219967993959602066523446427909, −7.41757584343701807322064534472, −6.01056497284882717171898989875, −5.13393042558710070424334107532, −4.25150378290994869253886744767, −3.42149505350259528480948610955, −1.66809823711069758352553434796, 1.80032994848751218277611598977, 2.73328193208534322920632797483, 3.49593670998182626944064256548, 5.34017943956434936016480260057, 6.13536402504083028551020953264, 6.98034932186204595045065503891, 7.84059179942768662685453308650, 9.265736037634512127238571628144, 9.664780801440130547760566958399, 10.92727288581322389834795420587

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