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Properties

Label 1-33-33.32-r0-0-0
Degree $1$
Conductor $33$
Sign $1$
Analytic cond. $0.153251$
Root an. cond. $0.153251$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s + 17-s − 19-s − 20-s − 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s − 43-s − 46-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s + 17-s − 19-s − 20-s − 23-s + 25-s − 26-s − 28-s + 29-s + 31-s + 32-s + 34-s + 35-s + 37-s − 38-s − 40-s + 41-s − 43-s − 46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(33\)    =    \(3 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.153251\)
Root analytic conductor: \(0.153251\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{33} (32, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 33,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.092291503\)
\(L(\frac12)\) \(\approx\) \(1.092291503\)
\(L(1)\) \(\approx\) \(1.332797188\)
\(L(1)\) \(\approx\) \(1.332797188\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + T \)
5 \( 1 - T \)
7 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 - T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 - T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−36.22670197160228655384957164617, −34.86979804370352953047987188318, −34.00052669826299603894318662317, −32.21508652100848086108238717144, −31.862053132219145424055672895195, −30.40972530504604648822877421808, −29.355507119523677931781543565610, −27.97065295685982701358440997141, −26.33665201440011595428737732531, −25.01194416737648581081374969192, −23.6330678372623130386463316360, −22.8032303504500431023975484460, −21.57562960541965370264093450500, −19.95148065570969446067032285389, −19.16857991604524510390831899173, −16.71025798128473101799872177954, −15.66217617484298026176809238344, −14.446117889281578588679046529210, −12.76111504593316981580874395970, −11.870654092261445677538594314170, −10.16870309638781907932950559899, −7.80326429364573470762626937960, −6.36152504478879300170016681928, −4.45916450759625444117471487542, −2.99695093746347198358466896022, 2.99695093746347198358466896022, 4.45916450759625444117471487542, 6.36152504478879300170016681928, 7.80326429364573470762626937960, 10.16870309638781907932950559899, 11.870654092261445677538594314170, 12.76111504593316981580874395970, 14.446117889281578588679046529210, 15.66217617484298026176809238344, 16.71025798128473101799872177954, 19.16857991604524510390831899173, 19.95148065570969446067032285389, 21.57562960541965370264093450500, 22.8032303504500431023975484460, 23.6330678372623130386463316360, 25.01194416737648581081374969192, 26.33665201440011595428737732531, 27.97065295685982701358440997141, 29.355507119523677931781543565610, 30.40972530504604648822877421808, 31.862053132219145424055672895195, 32.21508652100848086108238717144, 34.00052669826299603894318662317, 34.86979804370352953047987188318, 36.22670197160228655384957164617

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