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Properties

Label 2-546-39.5-c1-0-15
Degree $2$
Conductor $546$
Sign $-0.0753 + 0.997i$
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 + (−1.14 + 1.29i)3-s + 1.00i·4-s + (−0.237 − 0.237i)5-s + (1.72 − 0.103i)6-s + (−0.707 − 0.707i)7-s + (0.707 − 0.707i)8-s + (−0.359 − 2.97i)9-s + 0.335i·10-s + (−2.55 + 2.55i)11-s + (−1.29 − 1.14i)12-s + (3.37 − 1.25i)13-s + 1.00i·14-s + (0.579 − 0.0348i)15-s − 1.00·16-s − 5.26·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.663 + 0.748i)3-s + 0.500i·4-s + (−0.106 − 0.106i)5-s + (0.705 − 0.0424i)6-s + (−0.267 − 0.267i)7-s + (0.250 − 0.250i)8-s + (−0.119 − 0.992i)9-s + 0.106i·10-s + (−0.770 + 0.770i)11-s + (−0.374 − 0.331i)12-s + (0.937 − 0.348i)13-s + 0.267i·14-s + (0.149 − 0.00900i)15-s − 0.250·16-s − 1.27·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0753 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0753 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.0753 + 0.997i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.0753 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.370499 - 0.399558i\)
\(L(\frac12)\) \(\approx\) \(0.370499 - 0.399558i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.14 - 1.29i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-3.37 + 1.25i)T \)
good5 \( 1 + (0.237 + 0.237i)T + 5iT^{2} \)
11 \( 1 + (2.55 - 2.55i)T - 11iT^{2} \)
17 \( 1 + 5.26T + 17T^{2} \)
19 \( 1 + (-0.906 + 0.906i)T - 19iT^{2} \)
23 \( 1 - 1.48T + 23T^{2} \)
29 \( 1 + 7.36iT - 29T^{2} \)
31 \( 1 + (-7.20 + 7.20i)T - 31iT^{2} \)
37 \( 1 + (2.83 + 2.83i)T + 37iT^{2} \)
41 \( 1 + (1.32 + 1.32i)T + 41iT^{2} \)
43 \( 1 + 2.75iT - 43T^{2} \)
47 \( 1 + (-2.27 + 2.27i)T - 47iT^{2} \)
53 \( 1 + 8.80iT - 53T^{2} \)
59 \( 1 + (-0.785 + 0.785i)T - 59iT^{2} \)
61 \( 1 - 8.74T + 61T^{2} \)
67 \( 1 + (9.68 - 9.68i)T - 67iT^{2} \)
71 \( 1 + (4.94 + 4.94i)T + 71iT^{2} \)
73 \( 1 + (4.00 + 4.00i)T + 73iT^{2} \)
79 \( 1 + 8.50T + 79T^{2} \)
83 \( 1 + (-3.05 - 3.05i)T + 83iT^{2} \)
89 \( 1 + (1.62 - 1.62i)T - 89iT^{2} \)
97 \( 1 + (4.86 - 4.86i)T - 97iT^{2} \)
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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

−10.42078183257823165556076639448, −9.989335198591664145389015170890, −8.992548017785637105936867893133, −8.124491878797070747287475228102, −6.92197347126387142972506430933, −5.95519541496764291837108154912, −4.67039037818061404652706414710, −3.90955055440570089728108147012, −2.48305684795749131850799682636, −0.43168431612091270995873477246, 1.35925683665606030150394148884, 2.98966755512261068595778653511, 4.80651917043966929340461000764, 5.77689592318487573655867060111, 6.55923333098953770354752227752, 7.29547161903655632964326221257, 8.422458292363847381092540370567, 8.914454106207279876605255085192, 10.36681398498935520431700468659, 10.97528585403669041268196870444

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