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Properties

Label 2-1150-5.4-c3-0-20
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 + 3.26i·3-s − 4·4-s − 6.52·6-s + 27.7i·7-s − 8i·8-s + 16.3·9-s − 10.8·11-s − 13.0i·12-s − 36.9i·13-s − 55.5·14-s + 16·16-s + 118. i·17-s + 32.7i·18-s + 19.3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.627i·3-s − 0.5·4-s − 0.443·6-s + 1.50i·7-s − 0.353i·8-s + 0.606·9-s − 0.296·11-s − 0.313i·12-s − 0.788i·13-s − 1.06·14-s + 0.250·16-s + 1.69i·17-s + 0.428i·18-s + 0.233·19-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

\(L(2)\) \(\approx\) \(1.145468678\)
\(L(\frac12)\) \(\approx\) \(1.145468678\)
\(L(\frac{5}{2})\) 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 \( 1 - 3.26iT - 27T^{2} \)
7 \( 1 - 27.7iT - 343T^{2} \)
11 \( 1 + 10.8T + 1.33e3T^{2} \)
13 \( 1 + 36.9iT - 2.19e3T^{2} \)
17 \( 1 - 118. iT - 4.91e3T^{2} \)
19 \( 1 - 19.3T + 6.85e3T^{2} \)
29 \( 1 + 234.T + 2.43e4T^{2} \)
31 \( 1 - 165.T + 2.97e4T^{2} \)
37 \( 1 - 202. iT - 5.06e4T^{2} \)
41 \( 1 + 295.T + 6.89e4T^{2} \)
43 \( 1 - 65.9iT - 7.95e4T^{2} \)
47 \( 1 + 110. iT - 1.03e5T^{2} \)
53 \( 1 - 688. iT - 1.48e5T^{2} \)
59 \( 1 + 10.5T + 2.05e5T^{2} \)
61 \( 1 - 110.T + 2.26e5T^{2} \)
67 \( 1 + 643. iT - 3.00e5T^{2} \)
71 \( 1 - 143.T + 3.57e5T^{2} \)
73 \( 1 - 158. iT - 3.89e5T^{2} \)
79 \( 1 + 1.12e3T + 4.93e5T^{2} \)
83 \( 1 - 824. iT - 5.71e5T^{2} \)
89 \( 1 - 879.T + 7.04e5T^{2} \)
97 \( 1 + 938. iT - 9.12e5T^{2} \)
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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

−9.916354419504778235216396224947, −9.025547234367822423554073144268, −8.386129514893910103421364788732, −7.65118843390306201679487151414, −6.45411490972785111873474137133, −5.71725585726902576200707517373, −5.05615697864128135272055348087, −4.03343516623480237186590536186, −2.99126544011969767752453775947, −1.62541387133119907854911041620, 0.29182712784125083817976532661, 1.18859202185654549918554286718, 2.22241838118279713906820023562, 3.52143587810730615156515735524, 4.33542183295243527484280094864, 5.19853866983536930993984606885, 6.66194044243706489717245685817, 7.26261580627103117511068324779, 7.82928839486017808828843268169, 9.086212193674861018184865197565

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