Properties

Label 2-1305-1.1-c1-0-32
Degree $2$
Conductor $1305$
Sign $1$
Analytic cond. $10.4204$
Root an. cond. $3.22807$
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  + 2.23·2-s + 3.00·4-s + 5-s + 2·7-s + 2.23·8-s + 2.23·10-s + 2·11-s + 2·13-s + 4.47·14-s − 0.999·16-s − 4.47·17-s + 2·19-s + 3.00·20-s + 4.47·22-s + 2·23-s + 25-s + 4.47·26-s + 6.00·28-s + 29-s − 2·31-s − 6.70·32-s − 10.0·34-s + 2·35-s + 8.47·37-s + 4.47·38-s + 2.23·40-s − 2·41-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s + 0.447·5-s + 0.755·7-s + 0.790·8-s + 0.707·10-s + 0.603·11-s + 0.554·13-s + 1.19·14-s − 0.249·16-s − 1.08·17-s + 0.458·19-s + 0.670·20-s + 0.953·22-s + 0.417·23-s + 0.200·25-s + 0.877·26-s + 1.13·28-s + 0.185·29-s − 0.359·31-s − 1.18·32-s − 1.71·34-s + 0.338·35-s + 1.39·37-s + 0.725·38-s + 0.353·40-s − 0.312·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $1$
Analytic conductor: \(10.4204\)
Root analytic conductor: \(3.22807\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1305,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.704115535\)
\(L(\frac12)\) \(\approx\) \(4.704115535\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 - 2.23T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 8.47T + 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 8T + 59T^{2} \)
61 \( 1 - 6.94T + 61T^{2} \)
67 \( 1 - 2.94T + 67T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 - 3.52T + 73T^{2} \)
79 \( 1 + 2.94T + 79T^{2} \)
83 \( 1 - 14.9T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 17.4T + 97T^{2} \)
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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

−9.625900841454291880152396868980, −8.865484712951231010193715329900, −7.86197586207358495849286291673, −6.72586420824739866711297408280, −6.22909440419408126030761260107, −5.24663188247230810137767288229, −4.59712680392637223725403581272, −3.74441411913574919251521192013, −2.67225043140704148192645255234, −1.56534257372319950079568832720, 1.56534257372319950079568832720, 2.67225043140704148192645255234, 3.74441411913574919251521192013, 4.59712680392637223725403581272, 5.24663188247230810137767288229, 6.22909440419408126030761260107, 6.72586420824739866711297408280, 7.86197586207358495849286291673, 8.865484712951231010193715329900, 9.625900841454291880152396868980

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