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Properties

Label 2-1150-115.22-c1-0-2
Degree $2$
Conductor $1150$
Sign $-0.742 + 0.670i$
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 + (−1.62 + 1.62i)3-s + 1.00i·4-s − 2.30·6-s + (3.12 − 3.12i)7-s + (−0.707 + 0.707i)8-s − 2.30i·9-s − 3.07i·11-s + (−1.62 − 1.62i)12-s + (−3.74 + 3.74i)13-s + 4.41·14-s − 1.00·16-s + (−4.06 + 4.06i)17-s + (1.62 − 1.62i)18-s − 7.09·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.940 + 0.940i)3-s + 0.500i·4-s − 0.940·6-s + (1.18 − 1.18i)7-s + (−0.250 + 0.250i)8-s − 0.767i·9-s − 0.928i·11-s + (−0.470 − 0.470i)12-s + (−1.03 + 1.03i)13-s + 1.18·14-s − 0.250·16-s + (−0.986 + 0.986i)17-s + (0.383 − 0.383i)18-s − 1.62·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1150\)    =    \(2 \cdot 5^{2} \cdot 23\)
Sign: $-0.742 + 0.670i$
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: \((2,\ 1150,\ (\ :1/2),\ -0.742 + 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4598065997\)
\(L(\frac12)\) \(\approx\) \(0.4598065997\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.20 - 2.31i)T \)
good3 \( 1 + (1.62 - 1.62i)T - 3iT^{2} \)
7 \( 1 + (-3.12 + 3.12i)T - 7iT^{2} \)
11 \( 1 + 3.07iT - 11T^{2} \)
13 \( 1 + (3.74 - 3.74i)T - 13iT^{2} \)
17 \( 1 + (4.06 - 4.06i)T - 17iT^{2} \)
19 \( 1 + 7.09T + 19T^{2} \)
29 \( 1 + 2iT - 29T^{2} \)
31 \( 1 + 3.90T + 31T^{2} \)
37 \( 1 + (6.24 - 6.24i)T - 37iT^{2} \)
41 \( 1 + 4.30T + 41T^{2} \)
43 \( 1 + (-8.13 - 8.13i)T + 43iT^{2} \)
47 \( 1 + (-6.51 - 6.51i)T + 47iT^{2} \)
53 \( 1 + (4.35 + 4.35i)T + 53iT^{2} \)
59 \( 1 - 3.39iT - 59T^{2} \)
61 \( 1 + 3.07iT - 61T^{2} \)
67 \( 1 + (-3.78 + 3.78i)T - 67iT^{2} \)
71 \( 1 - 4.30T + 71T^{2} \)
73 \( 1 + (-0.986 + 0.986i)T - 73iT^{2} \)
79 \( 1 - 6.15T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (0.286 - 0.286i)T - 97iT^{2} \)
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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

−10.66411478252032528236559935277, −9.606555471731972640471499572520, −8.507907044886170738004888323732, −7.81106306315960617040765242363, −6.73689817966659016478993178262, −6.04617199499976650993298979691, −4.99009044013820177380993771781, −4.34350455368941527969811984890, −3.92830399464428210908716975987, −1.96055405326488551374763160936, 0.17445733770296326029142663286, 2.00802136063370285375204335087, 2.33100804374289310730677713546, 4.27722014370055791055374165608, 5.19894907013754800365350947225, 5.60470088217835950373844574946, 6.73349434236437534655877301694, 7.41334630179990836592842394022, 8.461062232542810064480896906447, 9.307426879238389278586809895425

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