Properties

Label 2-1150-5.4-c1-0-26
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 − 1.43i·3-s − 4-s + 1.43·6-s + 3.08i·7-s i·8-s + 0.950·9-s − 6.46·11-s + 1.43i·12-s − 3.95i·13-s − 3.08·14-s + 16-s − 3.43i·17-s + 0.950i·18-s − 3.08·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.826i·3-s − 0.5·4-s + 0.584·6-s + 1.16i·7-s − 0.353i·8-s + 0.316·9-s − 1.95·11-s + 0.413i·12-s − 1.09i·13-s − 0.825·14-s + 0.250·16-s − 0.832i·17-s + 0.224i·18-s − 0.708·19-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

\(L(1)\) \(\approx\) \(0.5245661030\)
\(L(\frac12)\) \(\approx\) \(0.5245661030\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.43iT - 3T^{2} \)
7 \( 1 - 3.08iT - 7T^{2} \)
11 \( 1 + 6.46T + 11T^{2} \)
13 \( 1 + 3.95iT - 13T^{2} \)
17 \( 1 + 3.43iT - 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
29 \( 1 + 0.863T + 29T^{2} \)
31 \( 1 + 5.95T + 31T^{2} \)
37 \( 1 + 7.03iT - 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 3.90iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 + 10.0iT - 67T^{2} \)
71 \( 1 - 2.56T + 71T^{2} \)
73 \( 1 + 5.90iT - 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 9.03iT - 83T^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−9.300457561933635429849668799000, −8.512724543714406314448906827277, −7.54778049521821499404863843087, −7.44508412799672235713284993068, −5.98693114029952959313923734529, −5.57778050711897890423261307286, −4.66043827352412813309012107324, −3.01182051812716714310107235149, −2.12572910393576819568297270070, −0.21796593097137714526795225984, 1.64778265089839964617259187778, 2.96239023006359401949048792511, 4.11973678190586253740737030984, 4.49262621326057561544454997090, 5.53436210961633479786582690392, 6.84158478020766761801411552888, 7.72725600707832388562184896869, 8.553374421385525067653688698620, 9.625876225215222728654414908039, 10.15949267786694263820499857127

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