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Properties

Label 2-1150-115.22-c1-0-14
Degree $2$
Conductor $1150$
Sign $0.450 - 0.892i$
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  + (−0.707 − 0.707i)2-s + (−2 + 2i)3-s + 1.00i·4-s + 2.82·6-s + (−2.61 + 2.61i)7-s + (0.707 − 0.707i)8-s − 5i·9-s − 0.317i·11-s + (−2.00 − 2.00i)12-s + (3.41 − 3.41i)13-s + 3.69·14-s − 1.00·16-s + (3.69 − 3.69i)17-s + (−3.53 + 3.53i)18-s + 5.09·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−1.15 + 1.15i)3-s + 0.500i·4-s + 1.15·6-s + (−0.987 + 0.987i)7-s + (0.250 − 0.250i)8-s − 1.66i·9-s − 0.0955i·11-s + (−0.577 − 0.577i)12-s + (0.946 − 0.946i)13-s + 0.987·14-s − 0.250·16-s + (0.896 − 0.896i)17-s + (−0.833 + 0.833i)18-s + 1.16·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(9.18279\)
Root analytic conductor: \(3.03031\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1150} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1150,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7199219803\)
\(L(\frac12)\) \(\approx\) \(0.7199219803\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
23 \( 1 + (-3.67 - 3.08i)T \)
good3 \( 1 + (2 - 2i)T - 3iT^{2} \)
7 \( 1 + (2.61 - 2.61i)T - 7iT^{2} \)
11 \( 1 + 0.317iT - 11T^{2} \)
13 \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \)
17 \( 1 + (-3.69 + 3.69i)T - 17iT^{2} \)
19 \( 1 - 5.09T + 19T^{2} \)
29 \( 1 - 3.65iT - 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 + (-2.07 + 2.07i)T - 37iT^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 + (-5.76 - 5.76i)T + 43iT^{2} \)
47 \( 1 + (1.58 + 1.58i)T + 47iT^{2} \)
53 \( 1 + (5.00 + 5.00i)T + 53iT^{2} \)
59 \( 1 + 9.65iT - 59T^{2} \)
61 \( 1 - 11.8iT - 61T^{2} \)
67 \( 1 + (2.38 - 2.38i)T - 67iT^{2} \)
71 \( 1 + 4.82T + 71T^{2} \)
73 \( 1 + (6.82 - 6.82i)T - 73iT^{2} \)
79 \( 1 - 4.59T + 79T^{2} \)
83 \( 1 + (-1.94 - 1.94i)T + 83iT^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 + (3.06 - 3.06i)T - 97iT^{2} \)
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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

−9.919568291202826670953168459797, −9.455888271000683424066349910011, −8.713262636326229389850381000451, −7.51866938239935496679691435072, −6.33932732688136617095086897483, −5.57044463159482107368911489687, −4.99120453473058575380971893462, −3.47337291423210478256884897371, −3.06479599746107270558597275251, −0.882384061681955572314345087586, 0.65881067148119718298267406416, 1.56727141020537744894257409936, 3.43351427155722960794540744416, 4.72535554201079391904521433680, 5.96608643939060144390369080381, 6.32422258298135870864572412812, 7.11951767045997638560891942333, 7.63956045206147157888245781379, 8.704189005050182925621131658464, 9.765062416469658185149711839407

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