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Properties

Label 2-546-21.17-c1-0-17
Degree $2$
Conductor $546$
Sign $0.948 + 0.317i$
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 + (1.62 − 0.588i)3-s + (0.499 − 0.866i)4-s + (−1.04 − 1.81i)5-s + (−1.11 + 1.32i)6-s + (1.84 + 1.89i)7-s + 0.999i·8-s + (2.30 − 1.91i)9-s + (1.81 + 1.04i)10-s + (1.67 + 0.965i)11-s + (0.305 − 1.70i)12-s + i·13-s + (−2.54 − 0.713i)14-s + (−2.77 − 2.33i)15-s + (−0.5 − 0.866i)16-s + (−0.345 + 0.599i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.940 − 0.339i)3-s + (0.249 − 0.433i)4-s + (−0.468 − 0.811i)5-s + (−0.455 + 0.540i)6-s + (0.699 + 0.715i)7-s + 0.353i·8-s + (0.769 − 0.638i)9-s + (0.573 + 0.331i)10-s + (0.504 + 0.291i)11-s + (0.0880 − 0.492i)12-s + 0.277i·13-s + (−0.680 − 0.190i)14-s + (−0.716 − 0.603i)15-s + (−0.125 − 0.216i)16-s + (−0.0839 + 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.948 + 0.317i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.948 + 0.317i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53014 - 0.249577i\)
\(L(\frac12)\) \(\approx\) \(1.53014 - 0.249577i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 + 0.588i)T \)
7 \( 1 + (-1.84 - 1.89i)T \)
13 \( 1 - iT \)
good5 \( 1 + (1.04 + 1.81i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.67 - 0.965i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.345 - 0.599i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.08 + 1.20i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.14 + 1.81i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.88iT - 29T^{2} \)
31 \( 1 + (-0.817 - 0.472i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.57 + 6.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.66T + 41T^{2} \)
43 \( 1 - 8.01T + 43T^{2} \)
47 \( 1 + (-2.19 - 3.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.81 + 2.77i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.21 - 9.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.92 + 1.68i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.98 - 8.62i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.58iT - 71T^{2} \)
73 \( 1 + (-7.46 - 4.31i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.33 + 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 + (-7.45 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.33iT - 97T^{2} \)
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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

−10.61852634231990174897655848987, −9.333723409313203762185680964459, −8.952804757870917064740625745687, −8.197872353172910439009095553698, −7.47231421876734214711665948617, −6.45513433872834680889021218807, −5.11265157885055363037797827477, −4.10067579224818029396295657725, −2.47699726426341713244533937965, −1.23138338313402153680174575409, 1.50446423158369363833249301342, 3.05550130093738793160785222362, 3.71394818137915044952774853098, 4.94827429798400136539220935149, 6.79581940469564071064871321154, 7.47796954728852599617299668848, 8.211269035143014367043662477415, 9.081932891170781976452838416740, 10.01302955809684864123129699594, 10.79227946407352120113592677561

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