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Properties

Label 2-1150-5.4-c3-0-46
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 − 7.73i·3-s − 4·4-s − 15.4·6-s + 23.5i·7-s + 8i·8-s − 32.7·9-s − 32.1·11-s + 30.9i·12-s − 40.0i·13-s + 47.1·14-s + 16·16-s + 126. i·17-s + 65.5i·18-s − 0.232·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.48i·3-s − 0.5·4-s − 1.05·6-s + 1.27i·7-s + 0.353i·8-s − 1.21·9-s − 0.880·11-s + 0.743i·12-s − 0.855i·13-s + 0.899·14-s + 0.250·16-s + 1.79i·17-s + 0.858i·18-s − 0.00281·19-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

\(L(2)\) \(\approx\) \(1.577355456\)
\(L(\frac12)\) \(\approx\) \(1.577355456\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.73iT - 27T^{2} \)
7 \( 1 - 23.5iT - 343T^{2} \)
11 \( 1 + 32.1T + 1.33e3T^{2} \)
13 \( 1 + 40.0iT - 2.19e3T^{2} \)
17 \( 1 - 126. iT - 4.91e3T^{2} \)
19 \( 1 + 0.232T + 6.85e3T^{2} \)
29 \( 1 - 137.T + 2.43e4T^{2} \)
31 \( 1 - 112.T + 2.97e4T^{2} \)
37 \( 1 - 45.7iT - 5.06e4T^{2} \)
41 \( 1 + 135.T + 6.89e4T^{2} \)
43 \( 1 + 543. iT - 7.95e4T^{2} \)
47 \( 1 - 26.4iT - 1.03e5T^{2} \)
53 \( 1 + 43.6iT - 1.48e5T^{2} \)
59 \( 1 + 202.T + 2.05e5T^{2} \)
61 \( 1 - 150.T + 2.26e5T^{2} \)
67 \( 1 + 420. iT - 3.00e5T^{2} \)
71 \( 1 - 667.T + 3.57e5T^{2} \)
73 \( 1 + 602. iT - 3.89e5T^{2} \)
79 \( 1 - 1.37e3T + 4.93e5T^{2} \)
83 \( 1 - 485. iT - 5.71e5T^{2} \)
89 \( 1 - 1.12e3T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3iT - 9.12e5T^{2} \)
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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

−8.938418775233899121198684584345, −8.233101931984030802797518943756, −7.82686099202815009154586973779, −6.53252105136262628979150766072, −5.83954432358902289199782951133, −5.04419826429258843965714622560, −3.45153086225936904641505547757, −2.44638239469383445074567032667, −1.83237039203859962639779073624, −0.59201565302198766998793916788, 0.69960945762109823618048636412, 2.81541567747125173488017819242, 3.85259887748155394407642596482, 4.73141028201800072733550553073, 5.01042866622004117027199130363, 6.38708657496571252618083711516, 7.22097570217136016359187972661, 8.013486599692207890421167111983, 9.034941239862368966112056846573, 9.729841907507800177642663640404

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