Properties

Label 2-1305-1.1-c1-0-38
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  + 2.43·2-s + 3.94·4-s + 5-s + 3.59·7-s + 4.75·8-s + 2.43·10-s − 6.58·11-s + 2.25·13-s + 8.77·14-s + 3.69·16-s − 0.256·17-s + 7.95·19-s + 3.94·20-s − 16.0·22-s − 0.961·23-s + 25-s + 5.50·26-s + 14.2·28-s − 29-s − 3.95·31-s − 0.498·32-s − 0.624·34-s + 3.59·35-s − 3.76·37-s + 19.3·38-s + 4.75·40-s − 5.00·41-s + ⋯
L(s)  = 1  + 1.72·2-s + 1.97·4-s + 0.447·5-s + 1.35·7-s + 1.67·8-s + 0.771·10-s − 1.98·11-s + 0.625·13-s + 2.34·14-s + 0.922·16-s − 0.0621·17-s + 1.82·19-s + 0.882·20-s − 3.42·22-s − 0.200·23-s + 0.200·25-s + 1.07·26-s + 2.68·28-s − 0.185·29-s − 0.709·31-s − 0.0881·32-s − 0.107·34-s + 0.607·35-s − 0.619·37-s + 3.14·38-s + 0.751·40-s − 0.781·41-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

\(L(1)\) \(\approx\) \(5.258789750\)
\(L(\frac12)\) \(\approx\) \(5.258789750\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.43T + 2T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
11 \( 1 + 6.58T + 11T^{2} \)
13 \( 1 - 2.25T + 13T^{2} \)
17 \( 1 + 0.256T + 17T^{2} \)
19 \( 1 - 7.95T + 19T^{2} \)
23 \( 1 + 0.961T + 23T^{2} \)
31 \( 1 + 3.95T + 31T^{2} \)
37 \( 1 + 3.76T + 37T^{2} \)
41 \( 1 + 5.00T + 41T^{2} \)
43 \( 1 + 6.19T + 43T^{2} \)
47 \( 1 - 7.02T + 47T^{2} \)
53 \( 1 + 6.38T + 53T^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 14.4T + 61T^{2} \)
67 \( 1 + 9.35T + 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 13.4T + 73T^{2} \)
79 \( 1 + 5.22T + 79T^{2} \)
83 \( 1 + 0.195T + 83T^{2} \)
89 \( 1 - 3.72T + 89T^{2} \)
97 \( 1 + 2.97T + 97T^{2} \)
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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

−9.971079346243260639257260814756, −8.605765780789077658496126119942, −7.74490129839161431734906149453, −7.09985597748738472235005163170, −5.80235997143234349569488823360, −5.26756342545988398399391876211, −4.85423144617327005746288550966, −3.60590845648355286468779699149, −2.67424995821009860596979784420, −1.67989400805514978999185844885, 1.67989400805514978999185844885, 2.67424995821009860596979784420, 3.60590845648355286468779699149, 4.85423144617327005746288550966, 5.26756342545988398399391876211, 5.80235997143234349569488823360, 7.09985597748738472235005163170, 7.74490129839161431734906149453, 8.605765780789077658496126119942, 9.971079346243260639257260814756

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