Properties

Label 2-74-1.1-c1-0-0
Degree $2$
Conductor $74$
Sign $1$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
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-s − 0.302·3-s + 4-s + 1.30·5-s + 0.302·6-s + 4.60·7-s − 8-s − 2.90·9-s − 1.30·10-s + 1.30·11-s − 0.302·12-s − 2.30·13-s − 4.60·14-s − 0.394·15-s + 16-s − 6·17-s + 2.90·18-s + 2·19-s + 1.30·20-s − 1.39·21-s − 1.30·22-s − 6.90·23-s + 0.302·24-s − 3.30·25-s + 2.30·26-s + 1.78·27-s + 4.60·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.174·3-s + 0.5·4-s + 0.582·5-s + 0.123·6-s + 1.74·7-s − 0.353·8-s − 0.969·9-s − 0.411·10-s + 0.392·11-s − 0.0874·12-s − 0.638·13-s − 1.23·14-s − 0.101·15-s + 0.250·16-s − 1.45·17-s + 0.685·18-s + 0.458·19-s + 0.291·20-s − 0.304·21-s − 0.277·22-s − 1.44·23-s + 0.0618·24-s − 0.660·25-s + 0.451·26-s + 0.344·27-s + 0.870·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $1$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7566511421\)
\(L(\frac12)\) \(\approx\) \(0.7566511421\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
37 \( 1 - T \)
good3 \( 1 + 0.302T + 3T^{2} \)
5 \( 1 - 1.30T + 5T^{2} \)
7 \( 1 - 4.60T + 7T^{2} \)
11 \( 1 - 1.30T + 11T^{2} \)
13 \( 1 + 2.30T + 13T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 6.90T + 23T^{2} \)
29 \( 1 - 6.90T + 29T^{2} \)
31 \( 1 - 3.30T + 31T^{2} \)
41 \( 1 + 0.908T + 41T^{2} \)
43 \( 1 + 6.60T + 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 - 14.5T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 8.69T + 73T^{2} \)
79 \( 1 + 16.1T + 79T^{2} \)
83 \( 1 - 17.2T + 83T^{2} \)
89 \( 1 - 5.21T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−14.49667430882702977079802065413, −13.82810602652582601657229089158, −11.90105752959434064909667781550, −11.33074162447793271165656847275, −10.11301099796250134694050439536, −8.769483072722598629061956387655, −7.900095471836613411558585840504, −6.27120580306579210087654420633, −4.84003718833468002241545765906, −2.09579133783652754251714950349, 2.09579133783652754251714950349, 4.84003718833468002241545765906, 6.27120580306579210087654420633, 7.900095471836613411558585840504, 8.769483072722598629061956387655, 10.11301099796250134694050439536, 11.33074162447793271165656847275, 11.90105752959434064909667781550, 13.82810602652582601657229089158, 14.49667430882702977079802065413

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