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Properties

Label 2-4650-5.4-c1-0-78
Degree $2$
Conductor $4650$
Sign $-0.447 + 0.894i$
Analytic cond. $37.1304$
Root an. cond. $6.09347$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s − 2i·7-s − i·8-s − 9-s − i·12-s − 4i·13-s + 2·14-s + 16-s + 6i·17-s − i·18-s + 2·21-s + 24-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.408·6-s − 0.755i·7-s − 0.353i·8-s − 0.333·9-s − 0.288i·12-s − 1.10i·13-s + 0.534·14-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.436·21-s + 0.204·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4650\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 31\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(37.1304\)
Root analytic conductor: \(6.09347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4650} (3349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 4650,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
31 \( 1 + T \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 8T + 29T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 14T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + 10iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 8iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 + 16T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−7.86798153457543047700327841859, −7.57205189343577426531811894959, −6.56955509588708589728637877685, −5.79394218739277014393196695718, −5.31637713066545321675048367050, −4.18686186716555405038559159172, −3.85734810567459559755141721949, −2.82270054810453001884704021596, −1.36539961704445881134697306256, 0, 1.37398900414161920287453066817, 2.26570765255069599326945272317, 2.89891066533009366071650719462, 3.96031087320868805595332593796, 4.80656365861342398591249533561, 5.59406535264662447531439259347, 6.31344918176610672302911855478, 7.23466254906777057130658716465, 7.76340854042030563405227955769

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