Properties

Label 2-403-1.1-c1-0-16
Degree $2$
Conductor $403$
Sign $1$
Analytic cond. $3.21797$
Root an. cond. $1.79387$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.14·2-s + 0.217·3-s + 2.58·4-s − 0.434·5-s + 0.464·6-s + 3.26·7-s + 1.24·8-s − 2.95·9-s − 0.931·10-s + 4.50·11-s + 0.560·12-s + 13-s + 6.98·14-s − 0.0944·15-s − 2.49·16-s − 3.22·17-s − 6.32·18-s + 0.614·19-s − 1.12·20-s + 0.708·21-s + 9.64·22-s + 3.65·23-s + 0.271·24-s − 4.81·25-s + 2.14·26-s − 1.29·27-s + 8.42·28-s + ⋯
L(s)  = 1  + 1.51·2-s + 0.125·3-s + 1.29·4-s − 0.194·5-s + 0.189·6-s + 1.23·7-s + 0.441·8-s − 0.984·9-s − 0.294·10-s + 1.35·11-s + 0.161·12-s + 0.277·13-s + 1.86·14-s − 0.0243·15-s − 0.623·16-s − 0.782·17-s − 1.48·18-s + 0.141·19-s − 0.251·20-s + 0.154·21-s + 2.05·22-s + 0.761·23-s + 0.0553·24-s − 0.962·25-s + 0.419·26-s − 0.248·27-s + 1.59·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(403\)    =    \(13 \cdot 31\)
Sign: $1$
Analytic conductor: \(3.21797\)
Root analytic conductor: \(1.79387\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 403,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.140884660\)
\(L(\frac12)\) \(\approx\) \(3.140884660\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 - T \)
31 \( 1 + T \)
good2 \( 1 - 2.14T + 2T^{2} \)
3 \( 1 - 0.217T + 3T^{2} \)
5 \( 1 + 0.434T + 5T^{2} \)
7 \( 1 - 3.26T + 7T^{2} \)
11 \( 1 - 4.50T + 11T^{2} \)
17 \( 1 + 3.22T + 17T^{2} \)
19 \( 1 - 0.614T + 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
37 \( 1 + 3.95T + 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 + 5.96T + 43T^{2} \)
47 \( 1 - 1.93T + 47T^{2} \)
53 \( 1 - 1.45T + 53T^{2} \)
59 \( 1 - 6.76T + 59T^{2} \)
61 \( 1 + 11.2T + 61T^{2} \)
67 \( 1 - 13.2T + 67T^{2} \)
71 \( 1 - 4.46T + 71T^{2} \)
73 \( 1 + 9.84T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 0.728T + 83T^{2} \)
89 \( 1 - 7.64T + 89T^{2} \)
97 \( 1 - 16.1T + 97T^{2} \)
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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

−11.44247791924643352501823070534, −11.05325639038714322555245721337, −9.243209790170937371584850029671, −8.513354638863638759658344062247, −7.27816391672180276239751036188, −6.19042316069268241050648659012, −5.31747099240211168952342441293, −4.32880461118389477482568694791, −3.45469306460464992156367271890, −1.97859814115490346250727288946, 1.97859814115490346250727288946, 3.45469306460464992156367271890, 4.32880461118389477482568694791, 5.31747099240211168952342441293, 6.19042316069268241050648659012, 7.27816391672180276239751036188, 8.513354638863638759658344062247, 9.243209790170937371584850029671, 11.05325639038714322555245721337, 11.44247791924643352501823070534

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