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Properties

Label 2-546-39.5-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.938 + 0.345i$
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 + (−1.68 − 0.398i)3-s + 1.00i·4-s + (−0.559 − 0.559i)5-s + (−0.909 − 1.47i)6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (2.68 + 1.34i)9-s − 0.790i·10-s + (−0.347 + 0.347i)11-s + (0.398 − 1.68i)12-s + (−3.45 + 1.03i)13-s − 1.00i·14-s + (0.719 + 1.16i)15-s − 1.00·16-s − 4.50·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.973 − 0.230i)3-s + 0.500i·4-s + (−0.250 − 0.250i)5-s + (−0.371 − 0.601i)6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.893 + 0.448i)9-s − 0.250i·10-s + (−0.104 + 0.104i)11-s + (0.115 − 0.486i)12-s + (−0.957 + 0.287i)13-s − 0.267i·14-s + (0.185 + 0.300i)15-s − 0.250·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.938 + 0.345i$
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: \((2,\ 546,\ (\ :1/2),\ -0.938 + 0.345i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0143963 - 0.0807478i\)
\(L(\frac12)\) \(\approx\) \(0.0143963 - 0.0807478i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.398i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (3.45 - 1.03i)T \)
good5 \( 1 + (0.559 + 0.559i)T + 5iT^{2} \)
11 \( 1 + (0.347 - 0.347i)T - 11iT^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
19 \( 1 + (3.86 - 3.86i)T - 19iT^{2} \)
23 \( 1 + 5.27T + 23T^{2} \)
29 \( 1 + 10.6iT - 29T^{2} \)
31 \( 1 + (2.47 - 2.47i)T - 31iT^{2} \)
37 \( 1 + (-6.92 - 6.92i)T + 37iT^{2} \)
41 \( 1 + (7.31 + 7.31i)T + 41iT^{2} \)
43 \( 1 - 7.24iT - 43T^{2} \)
47 \( 1 + (2.49 - 2.49i)T - 47iT^{2} \)
53 \( 1 - 12.0iT - 53T^{2} \)
59 \( 1 + (-3.41 + 3.41i)T - 59iT^{2} \)
61 \( 1 - 6.42T + 61T^{2} \)
67 \( 1 + (2.43 - 2.43i)T - 67iT^{2} \)
71 \( 1 + (8.49 + 8.49i)T + 71iT^{2} \)
73 \( 1 + (-5.93 - 5.93i)T + 73iT^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + (-6.31 - 6.31i)T + 83iT^{2} \)
89 \( 1 + (-12.2 + 12.2i)T - 89iT^{2} \)
97 \( 1 + (0.867 - 0.867i)T - 97iT^{2} \)
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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

−11.59324289082577406693936349979, −10.44169314913458702036140312452, −9.719999396761895076560208790247, −8.313408115916169676873407105170, −7.54543770849224369619575130953, −6.52257959776682093445355418511, −5.96799099098547357852551477143, −4.63723521303050033916306356681, −4.16524874781679756979139765656, −2.21201275067507488468171290073, 0.04198805029354520586007903800, 2.16210963548957773130978972429, 3.58725310104861181101118337713, 4.67056409391752907512784564372, 5.45543378573204449554634344686, 6.53505960729983760938938106442, 7.24839943552207467890975481636, 8.782238510032640044229218796566, 9.730884977438078703635106024445, 10.56315194645235783356922882510

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