Properties

Label 2-203-1.1-c1-0-0
Degree $2$
Conductor $203$
Sign $1$
Analytic cond. $1.62096$
Root an. cond. $1.27317$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s + 7-s − 2·9-s + 8·10-s + 2·11-s − 2·12-s + 4·13-s − 2·14-s + 4·15-s − 4·16-s − 2·17-s + 4·18-s + 5·19-s − 8·20-s − 21-s − 4·22-s + 9·23-s + 11·25-s − 8·26-s + 5·27-s + 2·28-s − 29-s − 8·30-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s + 0.377·7-s − 2/3·9-s + 2.52·10-s + 0.603·11-s − 0.577·12-s + 1.10·13-s − 0.534·14-s + 1.03·15-s − 16-s − 0.485·17-s + 0.942·18-s + 1.14·19-s − 1.78·20-s − 0.218·21-s − 0.852·22-s + 1.87·23-s + 11/5·25-s − 1.56·26-s + 0.962·27-s + 0.377·28-s − 0.185·29-s − 1.46·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 203 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(203\)    =    \(7 \cdot 29\)
Sign: $1$
Analytic conductor: \(1.62096\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 203,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3505417174\)
\(L(\frac12)\) \(\approx\) \(0.3505417174\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - T \)
29 \( 1 + T \)
good2 \( 1 + p T + p T^{2} \)
3 \( 1 + T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 - 7 T + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 3 T + p T^{2} \)
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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

−11.71095812379068084447954858997, −11.33602129559791134643564976933, −10.79131551663212647721316181454, −9.142061565933134180456079240781, −8.518787779003785407287218429384, −7.61819623880598528187301543377, −6.71652888402919591077959675143, −4.95184979272712106895823280322, −3.51539385012532972520182576397, −0.858748240911086982856794370077, 0.858748240911086982856794370077, 3.51539385012532972520182576397, 4.95184979272712106895823280322, 6.71652888402919591077959675143, 7.61819623880598528187301543377, 8.518787779003785407287218429384, 9.142061565933134180456079240781, 10.79131551663212647721316181454, 11.33602129559791134643564976933, 11.71095812379068084447954858997

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