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Properties

Label 2-546-39.5-c1-0-8
Degree $2$
Conductor $546$
Sign $-0.871 - 0.490i$
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 + (0.0462 + 1.73i)3-s + 1.00i·4-s + (1.43 + 1.43i)5-s + (−1.19 + 1.25i)6-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + (−2.99 + 0.160i)9-s + 2.02i·10-s + (−3.59 + 3.59i)11-s + (−1.73 + 0.0462i)12-s + (2.66 − 2.42i)13-s − 1.00i·14-s + (−2.41 + 2.54i)15-s − 1.00·16-s + 4.10·17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.0267 + 0.999i)3-s + 0.500i·4-s + (0.639 + 0.639i)5-s + (−0.486 + 0.513i)6-s + (−0.267 − 0.267i)7-s + (−0.250 + 0.250i)8-s + (−0.998 + 0.0533i)9-s + 0.639i·10-s + (−1.08 + 1.08i)11-s + (−0.499 + 0.0133i)12-s + (0.740 − 0.672i)13-s − 0.267i·14-s + (−0.622 + 0.656i)15-s − 0.250·16-s + 0.995·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.871 - 0.490i$
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: \((2,\ 546,\ (\ :1/2),\ -0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.458964 + 1.74958i\)
\(L(\frac12)\) \(\approx\) \(0.458964 + 1.74958i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0462 - 1.73i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (-2.66 + 2.42i)T \)
good5 \( 1 + (-1.43 - 1.43i)T + 5iT^{2} \)
11 \( 1 + (3.59 - 3.59i)T - 11iT^{2} \)
17 \( 1 - 4.10T + 17T^{2} \)
19 \( 1 + (4.90 - 4.90i)T - 19iT^{2} \)
23 \( 1 - 4.97T + 23T^{2} \)
29 \( 1 + 2.59iT - 29T^{2} \)
31 \( 1 + (1.00 - 1.00i)T - 31iT^{2} \)
37 \( 1 + (-8.01 - 8.01i)T + 37iT^{2} \)
41 \( 1 + (0.185 + 0.185i)T + 41iT^{2} \)
43 \( 1 + 8.21iT - 43T^{2} \)
47 \( 1 + (-2.56 + 2.56i)T - 47iT^{2} \)
53 \( 1 - 7.36iT - 53T^{2} \)
59 \( 1 + (1.33 - 1.33i)T - 59iT^{2} \)
61 \( 1 + 10.1T + 61T^{2} \)
67 \( 1 + (-3.13 + 3.13i)T - 67iT^{2} \)
71 \( 1 + (-9.44 - 9.44i)T + 71iT^{2} \)
73 \( 1 + (0.868 + 0.868i)T + 73iT^{2} \)
79 \( 1 - 12.6T + 79T^{2} \)
83 \( 1 + (0.232 + 0.232i)T + 83iT^{2} \)
89 \( 1 + (-3.93 + 3.93i)T - 89iT^{2} \)
97 \( 1 + (-7.65 + 7.65i)T - 97iT^{2} \)
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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

−10.76512406470347767194298425051, −10.35628225827445506262769083219, −9.654342237746511556888826336020, −8.398638424198864219203750211584, −7.59223830499440168884132041901, −6.34241056942918607677296865389, −5.63533147769995860859849383550, −4.65410603954663243697081909125, −3.53789987437974721624385976094, −2.53699426612400211211341772119, 0.907192332753740321577904903345, 2.27592070043299868185912835367, 3.29218744702471561976743790867, 4.96934647462956461433072815781, 5.79514623325726540666991855007, 6.48954906768893220571604515987, 7.78622741028536605557856213436, 8.802173967319558501697944036706, 9.366696309567614385654244183054, 10.88606439098740791459064902738

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