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Properties

Label 2-74-37.26-c1-0-2
Degree $2$
Conductor $74$
Sign $0.559 + 0.828i$
Analytic cond. $0.590892$
Root an. cond. $0.768695$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.258 + 0.447i)3-s + (−0.499 + 0.866i)4-s + (2.10 − 3.65i)5-s + 0.517·6-s + (0.258 − 0.447i)7-s + 0.999·8-s + (1.36 + 2.36i)9-s − 4.21·10-s − 3.73·11-s + (−0.258 − 0.447i)12-s + (−1 + 1.73i)13-s − 0.517·14-s + (1.09 + 1.88i)15-s + (−0.5 − 0.866i)16-s + (1.01 + 1.76i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.149 + 0.258i)3-s + (−0.249 + 0.433i)4-s + (0.942 − 1.63i)5-s + 0.211·6-s + (0.0977 − 0.169i)7-s + 0.353·8-s + (0.455 + 0.788i)9-s − 1.33·10-s − 1.12·11-s + (−0.0746 − 0.129i)12-s + (−0.277 + 0.480i)13-s − 0.138·14-s + (0.281 + 0.487i)15-s + (−0.125 − 0.216i)16-s + (0.246 + 0.427i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(74\)    =    \(2 \cdot 37\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(0.590892\)
Root analytic conductor: \(0.768695\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{74} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 74,\ (\ :1/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.730204 - 0.387870i\)
\(L(\frac12)\) \(\approx\) \(0.730204 - 0.387870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (4.10 + 4.48i)T \)
good3 \( 1 + (0.258 - 0.447i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.10 + 3.65i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.258 + 0.447i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.73T + 11T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.01 - 1.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.34 - 5.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.21T + 23T^{2} \)
29 \( 1 + 0.482T + 29T^{2} \)
31 \( 1 + 2.69T + 31T^{2} \)
41 \( 1 + (0.848 - 1.47i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 8.24T + 43T^{2} \)
47 \( 1 + 1.30T + 47T^{2} \)
53 \( 1 + (1.13 + 1.96i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.24 + 10.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.24 - 2.15i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.133 + 0.231i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.74 + 3.01i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 5.73T + 73T^{2} \)
79 \( 1 + (2.38 - 4.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.47 + 9.48i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.06 + 13.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.302T + 97T^{2} \)
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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

−14.06850383922288431820899501511, −12.92533783418393665808034756016, −12.58667242233411029729913030424, −10.83011851358173037811915521061, −9.972640011914718290371412207230, −8.917756248594326618278914076053, −7.81874030242604022224583846277, −5.52058422804668733797722400848, −4.50928289791509269796628937971, −1.83336077981812360404894545468, 2.75299454404612744510344940506, 5.40924915975832073907732961599, 6.67264220177622341610529616069, 7.36414970961526755457077853257, 9.181072141452219153811585329229, 10.27585521427037681088337984797, 11.07160223241842347738468169571, 12.87711468169208391164067963325, 13.80712313690247580603445719660, 15.09395254831850349162476314513

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