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Properties

Label 2-546-91.34-c1-0-13
Degree $2$
Conductor $546$
Sign $0.324 + 0.946i$
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 + i·3-s + 1.00i·4-s + (1.80 − 1.80i)5-s + (0.707 − 0.707i)6-s + (1.80 − 1.93i)7-s + (0.707 − 0.707i)8-s − 9-s − 2.54·10-s + (0.936 − 0.936i)11-s − 1.00·12-s + (−2 − 3i)13-s + (−2.64 + 0.0951i)14-s + (1.80 + 1.80i)15-s − 1.00·16-s + 0.496·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.577i·3-s + 0.500i·4-s + (0.806 − 0.806i)5-s + (0.288 − 0.288i)6-s + (0.681 − 0.732i)7-s + (0.250 − 0.250i)8-s − 0.333·9-s − 0.806·10-s + (0.282 − 0.282i)11-s − 0.288·12-s + (−0.554 − 0.832i)13-s + (−0.706 + 0.0254i)14-s + (0.465 + 0.465i)15-s − 0.250·16-s + 0.120·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.324 + 0.946i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.324 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05700 - 0.755251i\)
\(L(\frac12)\) \(\approx\) \(1.05700 - 0.755251i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-1.80 + 1.93i)T \)
13 \( 1 + (2 + 3i)T \)
good5 \( 1 + (-1.80 + 1.80i)T - 5iT^{2} \)
11 \( 1 + (-0.936 + 0.936i)T - 11iT^{2} \)
17 \( 1 - 0.496T + 17T^{2} \)
19 \( 1 + (3.07 - 3.07i)T - 19iT^{2} \)
23 \( 1 + 7.37iT - 23T^{2} \)
29 \( 1 - 7.25T + 29T^{2} \)
31 \( 1 + (5.73 - 5.73i)T - 31iT^{2} \)
37 \( 1 + (-3.54 + 3.54i)T - 37iT^{2} \)
41 \( 1 + (-6.11 + 6.11i)T - 41iT^{2} \)
43 \( 1 + 6.85iT - 43T^{2} \)
47 \( 1 + (-5.87 - 5.87i)T + 47iT^{2} \)
53 \( 1 + 4.26T + 53T^{2} \)
59 \( 1 + (-1.37 - 1.37i)T + 59iT^{2} \)
61 \( 1 + 0.496iT - 61T^{2} \)
67 \( 1 + (-9.11 - 9.11i)T + 67iT^{2} \)
71 \( 1 + (-11.3 - 11.3i)T + 71iT^{2} \)
73 \( 1 + (3.54 + 3.54i)T + 73iT^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 + (7.87 - 7.87i)T - 83iT^{2} \)
89 \( 1 + (10.2 + 10.2i)T + 89iT^{2} \)
97 \( 1 + (9.46 - 9.46i)T - 97iT^{2} \)
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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

−10.44840246391830323340293356961, −9.963278267969645280840011647415, −8.836176476003761413703476654414, −8.375853160252327243357933657857, −7.20805866606833058110779395337, −5.82479270476409935891476427142, −4.84776479149289346842588285738, −3.92649262030026455026508114785, −2.40089917919159525283055396152, −0.950748904173580360270601635977, 1.72344763683472812609571237772, 2.61976597307014987858378949513, 4.62075139034760871548220637317, 5.78108233566977583579278384213, 6.50190271566804942595345793731, 7.31833661884067078239183000038, 8.223145971290655903061159297455, 9.279068003162628333608900216864, 9.804938623061435057704460484986, 11.06523089530538540380813008759

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