Properties

Label 2-277350-1.1-c1-0-25
Degree $2$
Conductor $277350$
Sign $1$
Analytic cond. $2214.65$
Root an. cond. $47.0600$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s − 2·7-s + 8-s + 9-s − 3·11-s − 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s + 18-s − 19-s + 2·21-s − 3·22-s + 6·23-s − 24-s + 4·26-s − 27-s − 2·28-s + 6·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.436·21-s − 0.639·22-s + 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 277350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(277350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(2214.65\)
Root analytic conductor: \(47.0600\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{277350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 277350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.568394237\)
\(L(\frac12)\) \(\approx\) \(3.568394237\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
43 \( 1 \)
good7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−12.79785805268341, −12.34062947786968, −11.97516020407779, −11.41604221962382, −10.89672561902312, −10.53114503486702, −10.24972077100141, −9.663647372638623, −9.007337384151019, −8.615367216321567, −7.978749169794740, −7.558496622719962, −6.791984889395795, −6.652403616867320, −6.152226160059386, −5.500282875650248, −5.107291833781623, −4.878876405572604, −3.937490977834083, −3.460624132218885, −3.254822648860874, −2.426097900051108, −1.876747964133429, −1.011475190989102, −0.5355870156549921, 0.5355870156549921, 1.011475190989102, 1.876747964133429, 2.426097900051108, 3.254822648860874, 3.460624132218885, 3.937490977834083, 4.878876405572604, 5.107291833781623, 5.500282875650248, 6.152226160059386, 6.652403616867320, 6.791984889395795, 7.558496622719962, 7.978749169794740, 8.615367216321567, 9.007337384151019, 9.663647372638623, 10.24972077100141, 10.53114503486702, 10.89672561902312, 11.41604221962382, 11.97516020407779, 12.34062947786968, 12.79785805268341

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