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Properties

Label 2-1150-5.4-c3-0-51
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 + 0.589i·3-s − 4·4-s − 1.17·6-s + 18.5i·7-s − 8i·8-s + 26.6·9-s + 47.9·11-s − 2.35i·12-s + 42.3i·13-s − 37.0·14-s + 16·16-s − 1.70i·17-s + 53.3i·18-s − 21.4·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.113i·3-s − 0.5·4-s − 0.0802·6-s + 0.999i·7-s − 0.353i·8-s + 0.987·9-s + 1.31·11-s − 0.0567i·12-s + 0.903i·13-s − 0.706·14-s + 0.250·16-s − 0.0243i·17-s + 0.697i·18-s − 0.258·19-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

\(L(2)\) \(\approx\) \(2.453531310\)
\(L(\frac12)\) \(\approx\) \(2.453531310\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.589iT - 27T^{2} \)
7 \( 1 - 18.5iT - 343T^{2} \)
11 \( 1 - 47.9T + 1.33e3T^{2} \)
13 \( 1 - 42.3iT - 2.19e3T^{2} \)
17 \( 1 + 1.70iT - 4.91e3T^{2} \)
19 \( 1 + 21.4T + 6.85e3T^{2} \)
29 \( 1 + 57.6T + 2.43e4T^{2} \)
31 \( 1 - 295.T + 2.97e4T^{2} \)
37 \( 1 - 7.85iT - 5.06e4T^{2} \)
41 \( 1 - 465.T + 6.89e4T^{2} \)
43 \( 1 - 182. iT - 7.95e4T^{2} \)
47 \( 1 + 449. iT - 1.03e5T^{2} \)
53 \( 1 + 368. iT - 1.48e5T^{2} \)
59 \( 1 - 377.T + 2.05e5T^{2} \)
61 \( 1 - 849.T + 2.26e5T^{2} \)
67 \( 1 + 92.3iT - 3.00e5T^{2} \)
71 \( 1 + 626.T + 3.57e5T^{2} \)
73 \( 1 - 439. iT - 3.89e5T^{2} \)
79 \( 1 + 641.T + 4.93e5T^{2} \)
83 \( 1 + 609. iT - 5.71e5T^{2} \)
89 \( 1 + 1.12e3T + 7.04e5T^{2} \)
97 \( 1 - 1.42e3iT - 9.12e5T^{2} \)
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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

−9.469154946023111902872192512108, −8.902569645812813294655374238136, −8.113597192230950849152498172810, −6.93664370093997847839597434554, −6.52697866402448722593196527997, −5.56569325665969425353949590754, −4.48118742132648312696760975409, −3.87957334810724052990450028771, −2.32893257719290736609711920021, −1.11412862230001541476485473563, 0.74529133232682284213006923173, 1.40580023287752264160305148522, 2.81865737087088163279932728553, 4.05959195486291046681255154037, 4.32535530615805669323468892817, 5.74295023254039612086000205484, 6.77256846938113285112930540390, 7.49918632085029321058923975133, 8.394987768573654257755624199026, 9.433560101887238996405847723990

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