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Properties

Label 2-333270-1.1-c1-0-123
Degree $2$
Conductor $333270$
Sign $-1$
Analytic cond. $2661.17$
Root an. cond. $51.5865$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s + 4·13-s + 14-s + 16-s + 4·17-s + 2·19-s − 20-s + 25-s + 4·26-s + 28-s − 10·29-s − 6·31-s + 32-s + 4·34-s − 35-s + 6·37-s + 2·38-s − 40-s + 2·41-s + 4·43-s + 10·47-s + 49-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.458·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.188·28-s − 1.85·29-s − 1.07·31-s + 0.176·32-s + 0.685·34-s − 0.169·35-s + 0.986·37-s + 0.324·38-s − 0.158·40-s + 0.312·41-s + 0.609·43-s + 1.45·47-s + 1/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(333270\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(2661.17\)
Root analytic conductor: \(51.5865\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 333270,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 16 T + p T^{2} \)
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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

−12.84155645491776, −12.52051587135151, −11.81351083950439, −11.46544271363084, −11.15177092274993, −10.76699721715398, −10.19452899109375, −9.651798444929935, −9.067151808258251, −8.783580560928861, −7.988259775353545, −7.701815733174211, −7.354127300354405, −6.806466130148240, −6.022585278996763, −5.825611355801973, −5.330791137136743, −4.813278223022662, −4.112263294527556, −3.709933864068861, −3.480079274034617, −2.674369446029575, −2.151464062331086, −1.352119250658852, −1.012820901096817, 0, 1.012820901096817, 1.352119250658852, 2.151464062331086, 2.674369446029575, 3.480079274034617, 3.709933864068861, 4.112263294527556, 4.813278223022662, 5.330791137136743, 5.825611355801973, 6.022585278996763, 6.806466130148240, 7.354127300354405, 7.701815733174211, 7.988259775353545, 8.783580560928861, 9.067151808258251, 9.651798444929935, 10.19452899109375, 10.76699721715398, 11.15177092274993, 11.46544271363084, 11.81351083950439, 12.52051587135151, 12.84155645491776

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