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Properties

Label 2-546-91.31-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.889 + 0.457i$
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  + (−0.965 − 0.258i)2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.499i)4-s + (0.794 − 2.96i)5-s + (0.707 + 0.707i)6-s + (−1.32 − 2.29i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.53 + 2.66i)10-s + (5.70 − 1.52i)11-s + (−0.5 − 0.866i)12-s + (−2.68 − 2.40i)13-s + (0.684 + 2.55i)14-s + (−2.17 + 2.17i)15-s + (0.500 + 0.866i)16-s + (−2.42 + 4.19i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 + 0.249i)4-s + (0.355 − 1.32i)5-s + (0.288 + 0.288i)6-s + (−0.499 − 0.866i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.485 + 0.841i)10-s + (1.71 − 0.460i)11-s + (−0.144 − 0.249i)12-s + (−0.744 − 0.667i)13-s + (0.182 + 0.683i)14-s + (−0.560 + 0.560i)15-s + (0.125 + 0.216i)16-s + (−0.587 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.889 + 0.457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.889 + 0.457i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.889 + 0.457i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.178718 - 0.738795i\)
\(L(\frac12)\) \(\approx\) \(0.178718 - 0.738795i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.32 + 2.29i)T \)
13 \( 1 + (2.68 + 2.40i)T \)
good5 \( 1 + (-0.794 + 2.96i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-5.70 + 1.52i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.42 - 4.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.663 + 2.47i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.88 - 1.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 6.52T + 29T^{2} \)
31 \( 1 + (7.96 - 2.13i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.80 + 6.74i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.208 - 0.208i)T + 41iT^{2} \)
43 \( 1 + 4.79iT - 43T^{2} \)
47 \( 1 + (7.89 + 2.11i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.36 - 5.83i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.39 + 5.22i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (9.00 - 5.19i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.49 + 13.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-7.97 + 7.97i)T - 71iT^{2} \)
73 \( 1 + (-4.15 - 15.5i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.82 + 6.62i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.84 - 5.84i)T + 83iT^{2} \)
89 \( 1 + (-3.23 - 0.865i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-5.75 - 5.75i)T + 97iT^{2} \)
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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

−10.39715545879989762728434995715, −9.400552679567143290408152570667, −8.895219669451254511214060787625, −7.82894707732831782549366648648, −6.78826726126061142579863307634, −6.00862871904409356817721682081, −4.74637346875020254537939022582, −3.64807317833527496451943824999, −1.66535482219772811986089044391, −0.59464573858284912202261321101, 1.98567603726942304658818306749, 3.19249602910537556547991623299, 4.67650494233342031717585977967, 6.16271665857727864479382039657, 6.55808487649861425654126773847, 7.30432331371964782284627114542, 8.820943232443620402108943377715, 9.666171657097597868478591447115, 9.942544770002043046911549039759, 11.23890401989680540112409304600

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