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Properties

Label 2-1150-5.4-c1-0-2
Degree $2$
Conductor $1150$
Sign $-0.447 - 0.894i$
Analytic cond. $9.18279$
Root an. cond. $3.03031$
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  + i·2-s − 2.79i·3-s − 4-s + 2.79·6-s + 1.79i·7-s − i·8-s − 4.79·9-s − 0.791·11-s + 2.79i·12-s + 5.79i·13-s − 1.79·14-s + 16-s − 0.791i·17-s − 4.79i·18-s − 5.79·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.61i·3-s − 0.5·4-s + 1.13·6-s + 0.677i·7-s − 0.353i·8-s − 1.59·9-s − 0.238·11-s + 0.805i·12-s + 1.60i·13-s − 0.478·14-s + 0.250·16-s − 0.191i·17-s − 1.12i·18-s − 1.32·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (599, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6218593403\)
\(L(\frac12)\) \(\approx\) \(0.6218593403\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 + 2.79iT - 3T^{2} \)
7 \( 1 - 1.79iT - 7T^{2} \)
11 \( 1 + 0.791T + 11T^{2} \)
13 \( 1 - 5.79iT - 13T^{2} \)
17 \( 1 + 0.791iT - 17T^{2} \)
19 \( 1 + 5.79T + 19T^{2} \)
29 \( 1 + 7.58T + 29T^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 6.79T + 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 - 4.41iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 13.5T + 59T^{2} \)
61 \( 1 - 10.3T + 61T^{2} \)
67 \( 1 + 11.1iT - 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 12.7iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 - 7.95iT - 97T^{2} \)
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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

−9.746165839316470987697194404115, −8.886209482073451756815119225102, −8.313868021838498326249063757615, −7.43097414800608616025286537799, −6.74337858080463524270484211006, −6.19547543557998702462840041044, −5.28037637869802994592302022113, −4.06093002382884354079847168779, −2.47884923729181715286507339590, −1.59230662053104337268999333332, 0.26136752835545480426918554224, 2.33240231338201107967323795190, 3.65934856413330468657018806531, 3.90332644023144732613346490221, 5.13299169447405591550224827956, 5.62796808014673126560413713810, 7.14514637620332206045978255355, 8.318325934054135630180532819144, 8.853984321336949287945480380015, 9.960230862926583662586950184856

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