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Properties

Label 2-371-371.370-c0-0-1
Degree $2$
Conductor $371$
Sign $1$
Analytic cond. $0.185153$
Root an. cond. $0.430294$
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 + 7-s − 9-s − 2·11-s + 16-s − 25-s + 28-s + 2·29-s − 36-s − 2·37-s − 2·43-s − 2·44-s + 49-s + 53-s − 63-s + 64-s − 2·77-s + 81-s + 2·99-s − 100-s − 2·107-s + 112-s + 2·113-s + 2·116-s + ⋯
L(s)  = 1  + 4-s + 7-s − 9-s − 2·11-s + 16-s − 25-s + 28-s + 2·29-s − 36-s − 2·37-s − 2·43-s − 2·44-s + 49-s + 53-s − 63-s + 64-s − 2·77-s + 81-s + 2·99-s − 100-s − 2·107-s + 112-s + 2·113-s + 2·116-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 371 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(371\)    =    \(7 \cdot 53\)
Sign: $1$
Analytic conductor: \(0.185153\)
Root analytic conductor: \(0.430294\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{371} (370, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 371,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9508034376\)
\(L(\frac12)\) \(\approx\) \(0.9508034376\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
53 \( 1 - T \)
good2 \( ( 1 - T )( 1 + T ) \)
3 \( 1 + T^{2} \)
5 \( 1 + T^{2} \)
11 \( ( 1 + T )^{2} \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )^{2} \)
31 \( 1 + T^{2} \)
37 \( ( 1 + T )^{2} \)
41 \( 1 + T^{2} \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
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

−11.62479325603277528405148705480, −10.67207057662990713849300051086, −10.21407716149224239344755352182, −8.413114074606457076478965108052, −8.049423271083788546038699060131, −6.96397474590620092840590849066, −5.69087882281888234075190045219, −4.99800066892873196546342259182, −3.10749912315226981513581656920, −2.11132547647401970898770776530, 2.11132547647401970898770776530, 3.10749912315226981513581656920, 4.99800066892873196546342259182, 5.69087882281888234075190045219, 6.96397474590620092840590849066, 8.049423271083788546038699060131, 8.413114074606457076478965108052, 10.21407716149224239344755352182, 10.67207057662990713849300051086, 11.62479325603277528405148705480

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