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Properties

Label 2-1150-5.4-c3-0-49
Degree $2$
Conductor $1150$
Sign $0.447 - 0.894i$
Analytic cond. $67.8521$
Root an. cond. $8.23724$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 2.98i·3-s − 4·4-s + 5.97·6-s + 2.98i·7-s − 8i·8-s + 18.0·9-s + 68.6·11-s + 11.9i·12-s + 12.0i·13-s − 5.97·14-s + 16·16-s + 106. i·17-s + 36.1i·18-s + 48.4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.575i·3-s − 0.5·4-s + 0.406·6-s + 0.161i·7-s − 0.353i·8-s + 0.669·9-s + 1.88·11-s + 0.287i·12-s + 0.256i·13-s − 0.113·14-s + 0.250·16-s + 1.51i·17-s + 0.473i·18-s + 0.585·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(67.8521\)
Root analytic conductor: \(8.23724\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :3/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.415931543\)
\(L(\frac12)\) \(\approx\) \(2.415931543\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
23 \( 1 - 23iT \)
good3 \( 1 + 2.98iT - 27T^{2} \)
7 \( 1 - 2.98iT - 343T^{2} \)
11 \( 1 - 68.6T + 1.33e3T^{2} \)
13 \( 1 - 12.0iT - 2.19e3T^{2} \)
17 \( 1 - 106. iT - 4.91e3T^{2} \)
19 \( 1 - 48.4T + 6.85e3T^{2} \)
29 \( 1 + 135.T + 2.43e4T^{2} \)
31 \( 1 + 230.T + 2.97e4T^{2} \)
37 \( 1 + 107. iT - 5.06e4T^{2} \)
41 \( 1 - 394.T + 6.89e4T^{2} \)
43 \( 1 + 136. iT - 7.95e4T^{2} \)
47 \( 1 + 50.4iT - 1.03e5T^{2} \)
53 \( 1 - 414. iT - 1.48e5T^{2} \)
59 \( 1 - 183.T + 2.05e5T^{2} \)
61 \( 1 + 98.4T + 2.26e5T^{2} \)
67 \( 1 - 136. iT - 3.00e5T^{2} \)
71 \( 1 + 708.T + 3.57e5T^{2} \)
73 \( 1 - 689. iT - 3.89e5T^{2} \)
79 \( 1 + 546.T + 4.93e5T^{2} \)
83 \( 1 - 20.2iT - 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 + 1.11e3iT - 9.12e5T^{2} \)
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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

−9.224121195269456508562291662603, −8.848964724152867927168935197438, −7.63613742092845266110915793070, −7.14004752944100665829884686488, −6.27195905452521609808913582822, −5.67399465743575787929484883223, −4.20823668288099221134542361010, −3.75994373949286782253252079362, −1.89278883556974537204132910284, −1.05808425363997772378796888485, 0.71844524953528603692237871734, 1.73025246738755659651381313189, 3.16189881202455403661142440593, 3.96344155895369380810480016867, 4.66747367561750103443437862071, 5.69778692766597226475860256787, 6.90425712061264928542848096962, 7.55969093526478889339721429811, 8.989481056892829090418545706164, 9.349637538437446351850878908922

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