Properties

Degree $2$
Conductor $605$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3·3-s − 4-s + 5-s + 3·6-s − 3·7-s + 3·8-s + 6·9-s − 10-s + 3·12-s + 4·13-s + 3·14-s − 3·15-s − 16-s − 6·18-s + 4·19-s − 20-s + 9·21-s − 8·23-s − 9·24-s + 25-s − 4·26-s − 9·27-s + 3·28-s + 6·29-s + 3·30-s − 2·31-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.73·3-s − 1/2·4-s + 0.447·5-s + 1.22·6-s − 1.13·7-s + 1.06·8-s + 2·9-s − 0.316·10-s + 0.866·12-s + 1.10·13-s + 0.801·14-s − 0.774·15-s − 1/4·16-s − 1.41·18-s + 0.917·19-s − 0.223·20-s + 1.96·21-s − 1.66·23-s − 1.83·24-s + 1/5·25-s − 0.784·26-s − 1.73·27-s + 0.566·28-s + 1.11·29-s + 0.547·30-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{605} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 + T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−19.22218498439370, −18.46631195930457, −17.93721525029488, −17.56701124296190, −16.51192121660945, −16.28165881009908, −15.60081879489169, −13.90087108615112, −13.46568664491686, −12.50167859230518, −11.89040180292964, −10.77568974253887, −10.23018872792594, −9.639657197632512, −8.662767229663168, −7.417147561276069, −6.396018873635439, −5.834424233268130, −4.827969909357216, −3.661746819269322, −1.344128445284920, 0, 1.344128445284920, 3.661746819269322, 4.827969909357216, 5.834424233268130, 6.396018873635439, 7.417147561276069, 8.662767229663168, 9.639657197632512, 10.23018872792594, 10.77568974253887, 11.89040180292964, 12.50167859230518, 13.46568664491686, 13.90087108615112, 15.60081879489169, 16.28165881009908, 16.51192121660945, 17.56701124296190, 17.93721525029488, 18.46631195930457, 19.22218498439370

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