Properties

Label 2-1305-1.1-c1-0-21
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  + 1.82·2-s + 1.31·4-s + 5-s − 0.729·7-s − 1.24·8-s + 1.82·10-s − 0.729·11-s + 3.38·13-s − 1.32·14-s − 4.90·16-s + 5.74·17-s + 6.11·19-s + 1.31·20-s − 1.32·22-s + 9.48·23-s + 25-s + 6.15·26-s − 0.957·28-s + 29-s + 5.48·31-s − 6.42·32-s + 10.4·34-s − 0.729·35-s − 10.2·37-s + 11.1·38-s − 1.24·40-s + 11.3·41-s + ⋯
L(s)  = 1  + 1.28·2-s + 0.656·4-s + 0.447·5-s − 0.275·7-s − 0.441·8-s + 0.575·10-s − 0.219·11-s + 0.938·13-s − 0.354·14-s − 1.22·16-s + 1.39·17-s + 1.40·19-s + 0.293·20-s − 0.282·22-s + 1.97·23-s + 0.200·25-s + 1.20·26-s − 0.180·28-s + 0.185·29-s + 0.985·31-s − 1.13·32-s + 1.79·34-s − 0.123·35-s − 1.69·37-s + 1.80·38-s − 0.197·40-s + 1.76·41-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

\(L(1)\) \(\approx\) \(3.449252771\)
\(L(\frac12)\) \(\approx\) \(3.449252771\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.82T + 2T^{2} \)
7 \( 1 + 0.729T + 7T^{2} \)
11 \( 1 + 0.729T + 11T^{2} \)
13 \( 1 - 3.38T + 13T^{2} \)
17 \( 1 - 5.74T + 17T^{2} \)
19 \( 1 - 6.11T + 19T^{2} \)
23 \( 1 - 9.48T + 23T^{2} \)
31 \( 1 - 5.48T + 31T^{2} \)
37 \( 1 + 10.2T + 37T^{2} \)
41 \( 1 - 11.3T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 1.89T + 47T^{2} \)
53 \( 1 + 8.14T + 53T^{2} \)
59 \( 1 + 8.68T + 59T^{2} \)
61 \( 1 + 15.5T + 61T^{2} \)
67 \( 1 + 2.55T + 67T^{2} \)
71 \( 1 - 4.83T + 71T^{2} \)
73 \( 1 - 6.29T + 73T^{2} \)
79 \( 1 + 5.39T + 79T^{2} \)
83 \( 1 + 0.0848T + 83T^{2} \)
89 \( 1 + 4.63T + 89T^{2} \)
97 \( 1 - 1.30T + 97T^{2} \)
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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

−9.595574847194210332318906549945, −8.989283979204674971476991372227, −7.892762914151509364177961979057, −6.88022676623477708125217688283, −6.07866951918344313202125584355, −5.34064961383050596117241062739, −4.71050638380219598717202588461, −3.29358622387056447369279713306, −3.08369922037493253462869783834, −1.26988037798009943994208634503, 1.26988037798009943994208634503, 3.08369922037493253462869783834, 3.29358622387056447369279713306, 4.71050638380219598717202588461, 5.34064961383050596117241062739, 6.07866951918344313202125584355, 6.88022676623477708125217688283, 7.892762914151509364177961979057, 8.989283979204674971476991372227, 9.595574847194210332318906549945

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