Properties

Label 2-331-331.330-c0-0-1
Degree $2$
Conductor $331$
Sign $1$
Analytic cond. $0.165190$
Root an. cond. $0.406436$
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  + 4-s − 5-s + 9-s + 16-s − 17-s − 19-s − 20-s − 31-s + 36-s − 43-s − 45-s + 49-s − 53-s + 64-s − 67-s − 68-s − 71-s − 76-s − 79-s − 80-s + 81-s + 2·83-s + 85-s + 2·89-s + 95-s + 2·103-s − 109-s + ⋯
L(s)  = 1  + 4-s − 5-s + 9-s + 16-s − 17-s − 19-s − 20-s − 31-s + 36-s − 43-s − 45-s + 49-s − 53-s + 64-s − 67-s − 68-s − 71-s − 76-s − 79-s − 80-s + 81-s + 2·83-s + 85-s + 2·89-s + 95-s + 2·103-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(331\)
Sign: $1$
Analytic conductor: \(0.165190\)
Root analytic conductor: \(0.406436\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{331} (330, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 331,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8602296346\)
\(L(\frac12)\) \(\approx\) \(0.8602296346\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad331 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 + T + T^{2} \)
7 \( ( 1 - T )( 1 + T ) \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( 1 + T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( 1 + T + T^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( 1 + T + T^{2} \)
83 \( ( 1 - T )^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−11.74292826314796643656111275461, −10.95237445072461570525508844139, −10.23266083055183576207708023595, −8.903706172792838095372521708606, −7.79180940520932362394377395805, −7.10385284471483772626094148120, −6.20835629064935385794746843377, −4.60474706744117454383959704076, −3.57222521465823825976349946309, −1.98816870243366142600134302603, 1.98816870243366142600134302603, 3.57222521465823825976349946309, 4.60474706744117454383959704076, 6.20835629064935385794746843377, 7.10385284471483772626094148120, 7.79180940520932362394377395805, 8.903706172792838095372521708606, 10.23266083055183576207708023595, 10.95237445072461570525508844139, 11.74292826314796643656111275461

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