Properties

Label 2-1305-1.1-c3-0-110
Degree $2$
Conductor $1305$
Sign $-1$
Analytic cond. $76.9974$
Root an. cond. $8.77482$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 17·4-s − 5·5-s + 16·7-s − 45·8-s + 25·10-s + 44·11-s + 78·13-s − 80·14-s + 89·16-s − 18·17-s − 28·19-s − 85·20-s − 220·22-s − 184·23-s + 25·25-s − 390·26-s + 272·28-s − 29·29-s − 224·31-s − 85·32-s + 90·34-s − 80·35-s + 254·37-s + 140·38-s + 225·40-s + 78·41-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s − 0.447·5-s + 0.863·7-s − 1.98·8-s + 0.790·10-s + 1.20·11-s + 1.66·13-s − 1.52·14-s + 1.39·16-s − 0.256·17-s − 0.338·19-s − 0.950·20-s − 2.13·22-s − 1.66·23-s + 1/5·25-s − 2.94·26-s + 1.83·28-s − 0.185·29-s − 1.29·31-s − 0.469·32-s + 0.453·34-s − 0.386·35-s + 1.12·37-s + 0.597·38-s + 0.889·40-s + 0.297·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1305\)    =    \(3^{2} \cdot 5 \cdot 29\)
Sign: $-1$
Analytic conductor: \(76.9974\)
Root analytic conductor: \(8.77482\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1305,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + p T \)
29 \( 1 + p T \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
7 \( 1 - 16 T + p^{3} T^{2} \)
11 \( 1 - 4 p T + p^{3} T^{2} \)
13 \( 1 - 6 p T + p^{3} T^{2} \)
17 \( 1 + 18 T + p^{3} T^{2} \)
19 \( 1 + 28 T + p^{3} T^{2} \)
23 \( 1 + 8 p T + p^{3} T^{2} \)
31 \( 1 + 224 T + p^{3} T^{2} \)
37 \( 1 - 254 T + p^{3} T^{2} \)
41 \( 1 - 78 T + p^{3} T^{2} \)
43 \( 1 + 260 T + p^{3} T^{2} \)
47 \( 1 + 312 T + p^{3} T^{2} \)
53 \( 1 + 574 T + p^{3} T^{2} \)
59 \( 1 + 180 T + p^{3} T^{2} \)
61 \( 1 + 10 p T + p^{3} T^{2} \)
67 \( 1 + 340 T + p^{3} T^{2} \)
71 \( 1 + 296 T + p^{3} T^{2} \)
73 \( 1 - 394 T + p^{3} T^{2} \)
79 \( 1 + 960 T + p^{3} T^{2} \)
83 \( 1 - 908 T + p^{3} T^{2} \)
89 \( 1 - 990 T + p^{3} T^{2} \)
97 \( 1 - 1234 T + p^{3} T^{2} \)
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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

−8.863797840009503943943455283502, −8.131183534365953856200451292880, −7.69924303482950387301903155407, −6.54800081437755376337741164995, −6.04983983804566913028569760101, −4.41558186204933310566281906471, −3.46804834126453856833586438062, −1.85050503471516195393091753506, −1.31185926918867997148460769775, 0, 1.31185926918867997148460769775, 1.85050503471516195393091753506, 3.46804834126453856833586438062, 4.41558186204933310566281906471, 6.04983983804566913028569760101, 6.54800081437755376337741164995, 7.69924303482950387301903155407, 8.131183534365953856200451292880, 8.863797840009503943943455283502

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