Properties

Label 2-1150-5.4-c3-0-3
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 − 4.74i·3-s − 4·4-s + 9.49·6-s − 29.3i·7-s − 8i·8-s + 4.44·9-s − 38.1·11-s + 18.9i·12-s − 22.5i·13-s + 58.7·14-s + 16·16-s + 104. i·17-s + 8.89i·18-s − 141.·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.913i·3-s − 0.5·4-s + 0.646·6-s − 1.58i·7-s − 0.353i·8-s + 0.164·9-s − 1.04·11-s + 0.456i·12-s − 0.480i·13-s + 1.12·14-s + 0.250·16-s + 1.48i·17-s + 0.116i·18-s − 1.71·19-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

\(L(2)\) \(\approx\) \(0.2435851178\)
\(L(\frac12)\) \(\approx\) \(0.2435851178\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.74iT - 27T^{2} \)
7 \( 1 + 29.3iT - 343T^{2} \)
11 \( 1 + 38.1T + 1.33e3T^{2} \)
13 \( 1 + 22.5iT - 2.19e3T^{2} \)
17 \( 1 - 104. iT - 4.91e3T^{2} \)
19 \( 1 + 141.T + 6.85e3T^{2} \)
29 \( 1 + 241.T + 2.43e4T^{2} \)
31 \( 1 - 99.2T + 2.97e4T^{2} \)
37 \( 1 + 59.9iT - 5.06e4T^{2} \)
41 \( 1 - 249.T + 6.89e4T^{2} \)
43 \( 1 + 163. iT - 7.95e4T^{2} \)
47 \( 1 - 205. iT - 1.03e5T^{2} \)
53 \( 1 - 491. iT - 1.48e5T^{2} \)
59 \( 1 + 433.T + 2.05e5T^{2} \)
61 \( 1 - 660.T + 2.26e5T^{2} \)
67 \( 1 - 323. iT - 3.00e5T^{2} \)
71 \( 1 - 893.T + 3.57e5T^{2} \)
73 \( 1 - 196. iT - 3.89e5T^{2} \)
79 \( 1 - 500.T + 4.93e5T^{2} \)
83 \( 1 - 800. iT - 5.71e5T^{2} \)
89 \( 1 - 729.T + 7.04e5T^{2} \)
97 \( 1 + 1.13e3iT - 9.12e5T^{2} \)
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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

−9.730882908076028824722682872370, −8.390868392663685600426997662253, −7.88145733873188362345653012695, −7.26973166335987193774360735405, −6.51550640811371884531618863074, −5.74890293314414450475458580775, −4.43686054349442100038109914231, −3.82230631453057098115959346987, −2.18753232541448796712264577026, −0.978073184713639528164462533027, 0.06567471794071414615777394119, 2.02321700426243220615038303503, 2.69216156754354786340274934855, 3.82894611157484328391020014665, 4.88376055798494515836811788501, 5.32915218728611544938779293053, 6.47563205221732268201684172268, 7.76789336624299573276810649644, 8.702364419353411016194709307799, 9.349466699447096663107953247402

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