Properties

Label 2-1610-1.1-c1-0-7
Degree $2$
Conductor $1610$
Sign $1$
Analytic cond. $12.8559$
Root an. cond. $3.58551$
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-s − 2·3-s + 4-s − 5-s − 2·6-s + 7-s + 8-s + 9-s − 10-s − 6·11-s − 2·12-s − 4·13-s + 14-s + 2·15-s + 16-s + 6·17-s + 18-s + 8·19-s − 20-s − 2·21-s − 6·22-s + 23-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.80·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.83·19-s − 0.223·20-s − 0.436·21-s − 1.27·22-s + 0.208·23-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1610\)    =    \(2 \cdot 5 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(12.8559\)
Root analytic conductor: \(3.58551\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1610,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.417364992\)
\(L(\frac12)\) \(\approx\) \(1.417364992\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 - T \)
good3 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.804138627097304869465745836991, −8.285746504554624384872788431081, −7.55884584235795011657377627575, −7.05520108669387958265164424169, −5.72447800456945246360797480173, −5.21022855726840300708586043221, −4.89123141179020227048767856254, −3.42366599428342128168538377727, −2.55080738571920609620281269689, −0.78351974673693313575593831890, 0.78351974673693313575593831890, 2.55080738571920609620281269689, 3.42366599428342128168538377727, 4.89123141179020227048767856254, 5.21022855726840300708586043221, 5.72447800456945246360797480173, 7.05520108669387958265164424169, 7.55884584235795011657377627575, 8.285746504554624384872788431081, 9.804138627097304869465745836991

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