Properties

Label 2-370-1.1-c1-0-6
Degree $2$
Conductor $370$
Sign $1$
Analytic cond. $2.95446$
Root an. cond. $1.71885$
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 + 2·3-s + 4-s + 5-s + 2·6-s − 4.37·7-s + 8-s + 9-s + 10-s + 2.37·11-s + 2·12-s + 6.74·13-s − 4.37·14-s + 2·15-s + 16-s + 0.372·17-s + 18-s − 2·19-s + 20-s − 8.74·21-s + 2.37·22-s − 4.74·23-s + 2·24-s + 25-s + 6.74·26-s − 4·27-s − 4.37·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 0.5·4-s + 0.447·5-s + 0.816·6-s − 1.65·7-s + 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.715·11-s + 0.577·12-s + 1.87·13-s − 1.16·14-s + 0.516·15-s + 0.250·16-s + 0.0902·17-s + 0.235·18-s − 0.458·19-s + 0.223·20-s − 1.90·21-s + 0.505·22-s − 0.989·23-s + 0.408·24-s + 0.200·25-s + 1.32·26-s − 0.769·27-s − 0.826·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370\)    =    \(2 \cdot 5 \cdot 37\)
Sign: $1$
Analytic conductor: \(2.95446\)
Root analytic conductor: \(1.71885\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{370} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 370,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.710486559\)
\(L(\frac12)\) \(\approx\) \(2.710486559\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 - T \)
37 \( 1 - T \)
good3 \( 1 - 2T + 3T^{2} \)
7 \( 1 + 4.37T + 7T^{2} \)
11 \( 1 - 2.37T + 11T^{2} \)
13 \( 1 - 6.74T + 13T^{2} \)
17 \( 1 - 0.372T + 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 + 4.74T + 23T^{2} \)
29 \( 1 + 9.11T + 29T^{2} \)
31 \( 1 + 8.37T + 31T^{2} \)
41 \( 1 + 0.372T + 41T^{2} \)
43 \( 1 - 1.62T + 43T^{2} \)
47 \( 1 - 2.74T + 47T^{2} \)
53 \( 1 + 4.37T + 53T^{2} \)
59 \( 1 - 1.25T + 59T^{2} \)
61 \( 1 - 0.372T + 61T^{2} \)
67 \( 1 + 6.74T + 67T^{2} \)
71 \( 1 - 4.74T + 71T^{2} \)
73 \( 1 + 2.74T + 73T^{2} \)
79 \( 1 - 6.74T + 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 17.1T + 97T^{2} \)
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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

−11.46613437349725801106723157171, −10.43111387338297534143893299370, −9.312317635453180027617770363635, −8.895229980931036036550921747096, −7.57899665809056725753882985405, −6.34714050316419728421997080073, −5.87169576061093445842092703018, −3.74786059982972828851228147626, −3.49455304925544319783485761219, −2.02259618606569259686588845236, 2.02259618606569259686588845236, 3.49455304925544319783485761219, 3.74786059982972828851228147626, 5.87169576061093445842092703018, 6.34714050316419728421997080073, 7.57899665809056725753882985405, 8.895229980931036036550921747096, 9.312317635453180027617770363635, 10.43111387338297534143893299370, 11.46613437349725801106723157171

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