Properties

Label 2-1305-1.1-c1-0-35
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  + 2.57·2-s + 4.63·4-s + 5-s − 2.24·7-s + 6.77·8-s + 2.57·10-s + 3.75·11-s − 1.55·13-s − 5.78·14-s + 8.18·16-s + 3.55·17-s − 1.18·19-s + 4.63·20-s + 9.67·22-s + 3.67·23-s + 25-s − 3.99·26-s − 10.4·28-s − 29-s + 5.18·31-s + 7.52·32-s + 9.14·34-s − 2.24·35-s + 0.969·37-s − 3.06·38-s + 6.77·40-s − 11.3·41-s + ⋯
L(s)  = 1  + 1.82·2-s + 2.31·4-s + 0.447·5-s − 0.849·7-s + 2.39·8-s + 0.814·10-s + 1.13·11-s − 0.430·13-s − 1.54·14-s + 2.04·16-s + 0.861·17-s − 0.272·19-s + 1.03·20-s + 2.06·22-s + 0.767·23-s + 0.200·25-s − 0.783·26-s − 1.96·28-s − 0.185·29-s + 0.931·31-s + 1.32·32-s + 1.56·34-s − 0.380·35-s + 0.159·37-s − 0.496·38-s + 1.07·40-s − 1.77·41-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

\(L(1)\) \(\approx\) \(5.262378274\)
\(L(\frac12)\) \(\approx\) \(5.262378274\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.57T + 2T^{2} \)
7 \( 1 + 2.24T + 7T^{2} \)
11 \( 1 - 3.75T + 11T^{2} \)
13 \( 1 + 1.55T + 13T^{2} \)
17 \( 1 - 3.55T + 17T^{2} \)
19 \( 1 + 1.18T + 19T^{2} \)
23 \( 1 - 3.67T + 23T^{2} \)
31 \( 1 - 5.18T + 31T^{2} \)
37 \( 1 - 0.969T + 37T^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + 0.884T + 43T^{2} \)
47 \( 1 + 11.2T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 + 8.49T + 59T^{2} \)
61 \( 1 + 10.8T + 61T^{2} \)
67 \( 1 + 4.05T + 67T^{2} \)
71 \( 1 + 6.37T + 71T^{2} \)
73 \( 1 + 3.26T + 73T^{2} \)
79 \( 1 - 6.32T + 79T^{2} \)
83 \( 1 - 9.03T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−9.852872886597645732785729045592, −8.943162569605366770590729745545, −7.61402848455396298250783448301, −6.60194621418094412216583396668, −6.35858545593319815245247592119, −5.33842305337891146467284011456, −4.56669326230324690579875634855, −3.52328997585583340489255792735, −2.91873155956535377979951521915, −1.60844290412381499215295695601, 1.60844290412381499215295695601, 2.91873155956535377979951521915, 3.52328997585583340489255792735, 4.56669326230324690579875634855, 5.33842305337891146467284011456, 6.35858545593319815245247592119, 6.60194621418094412216583396668, 7.61402848455396298250783448301, 8.943162569605366770590729745545, 9.852872886597645732785729045592

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