Properties

Label 2-1305-1.1-c1-0-26
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  + 2.41·2-s + 3.82·4-s − 5-s + 0.828·7-s + 4.41·8-s − 2.41·10-s + 4.82·11-s − 2·13-s + 1.99·14-s + 2.99·16-s + 2.82·17-s + 0.828·19-s − 3.82·20-s + 11.6·22-s + 8.82·23-s + 25-s − 4.82·26-s + 3.17·28-s − 29-s − 10.4·31-s − 1.58·32-s + 6.82·34-s − 0.828·35-s + 8.48·37-s + 1.99·38-s − 4.41·40-s + 6·41-s + ⋯
L(s)  = 1  + 1.70·2-s + 1.91·4-s − 0.447·5-s + 0.313·7-s + 1.56·8-s − 0.763·10-s + 1.45·11-s − 0.554·13-s + 0.534·14-s + 0.749·16-s + 0.685·17-s + 0.190·19-s − 0.856·20-s + 2.48·22-s + 1.84·23-s + 0.200·25-s − 0.946·26-s + 0.599·28-s − 0.185·29-s − 1.88·31-s − 0.280·32-s + 1.17·34-s − 0.140·35-s + 1.39·37-s + 0.324·38-s − 0.697·40-s + 0.937·41-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

\(L(1)\) \(\approx\) \(4.621058995\)
\(L(\frac12)\) \(\approx\) \(4.621058995\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
7 \( 1 - 0.828T + 7T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 2.82T + 17T^{2} \)
19 \( 1 - 0.828T + 19T^{2} \)
23 \( 1 - 8.82T + 23T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 8.48T + 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 7.65T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 7.65T + 61T^{2} \)
67 \( 1 + 10.4T + 67T^{2} \)
71 \( 1 + 7.31T + 71T^{2} \)
73 \( 1 + 8.48T + 73T^{2} \)
79 \( 1 - 14.4T + 79T^{2} \)
83 \( 1 + 12.8T + 83T^{2} \)
89 \( 1 + 3.65T + 89T^{2} \)
97 \( 1 - 4.48T + 97T^{2} \)
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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

−9.631174227195902624716691172400, −8.886154431504759361613417113661, −7.58108352477538053422413409652, −7.02024890940965534265027683822, −6.14258378482332501032300726916, −5.24682186013868870229665504158, −4.50621496327616943337644256350, −3.68695708865252579033773835856, −2.90236261582425638504990454802, −1.47727387568019680231634409241, 1.47727387568019680231634409241, 2.90236261582425638504990454802, 3.68695708865252579033773835856, 4.50621496327616943337644256350, 5.24682186013868870229665504158, 6.14258378482332501032300726916, 7.02024890940965534265027683822, 7.58108352477538053422413409652, 8.886154431504759361613417113661, 9.631174227195902624716691172400

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