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Properties

Label 2-1232-308.307-c0-0-0
Degree $2$
Conductor $1232$
Sign $1$
Analytic cond. $0.614848$
Root an. cond. $0.784122$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·3-s − 7-s + 1.00·9-s + 11-s − 1.41·13-s + 1.41·17-s + 1.41·21-s + 25-s + 1.41·31-s − 1.41·33-s + 2.00·39-s − 1.41·41-s + 1.41·47-s + 49-s − 2.00·51-s + 1.41·59-s + 1.41·61-s − 1.00·63-s + 1.41·73-s − 1.41·75-s − 77-s − 0.999·81-s + 1.41·91-s − 2.00·93-s + 1.00·99-s − 1.41·101-s − 1.41·103-s + ⋯
L(s)  = 1  − 1.41·3-s − 7-s + 1.00·9-s + 11-s − 1.41·13-s + 1.41·17-s + 1.41·21-s + 25-s + 1.41·31-s − 1.41·33-s + 2.00·39-s − 1.41·41-s + 1.41·47-s + 49-s − 2.00·51-s + 1.41·59-s + 1.41·61-s − 1.00·63-s + 1.41·73-s − 1.41·75-s − 77-s − 0.999·81-s + 1.41·91-s − 2.00·93-s + 1.00·99-s − 1.41·101-s − 1.41·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1232\)    =    \(2^{4} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(0.614848\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1232} (1231, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1232,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5905913652\)
\(L(\frac12)\) \(\approx\) \(0.5905913652\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + T \)
11 \( 1 - T \)
good3 \( 1 + 1.41T + T^{2} \)
5 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - 1.41T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.41T + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.41T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.989884654535904609838220931096, −9.438524914006699872709974138389, −8.255719782207309413574074233588, −6.95872131924122525066854052081, −6.71866836355569930413309302661, −5.67324070808647977646971227387, −5.04985741891000603286807182279, −3.95003603123827226792794163141, −2.76616249288993126104631806494, −0.919856199549772509601029235173, 0.919856199549772509601029235173, 2.76616249288993126104631806494, 3.95003603123827226792794163141, 5.04985741891000603286807182279, 5.67324070808647977646971227387, 6.71866836355569930413309302661, 6.95872131924122525066854052081, 8.255719782207309413574074233588, 9.438524914006699872709974138389, 9.989884654535904609838220931096

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