Properties

Label 2-930-1.1-c1-0-7
Degree $2$
Conductor $930$
Sign $1$
Analytic cond. $7.42608$
Root an. cond. $2.72508$
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 − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s + 5·11-s − 12-s − 6·13-s + 3·14-s + 15-s + 16-s − 4·17-s + 18-s + 5·19-s − 20-s − 3·21-s + 5·22-s + 5·23-s − 24-s + 25-s − 6·26-s − 27-s + 3·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.50·11-s − 0.288·12-s − 1.66·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 1.14·19-s − 0.223·20-s − 0.654·21-s + 1.06·22-s + 1.04·23-s − 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(930\)    =    \(2 \cdot 3 \cdot 5 \cdot 31\)
Sign: $1$
Analytic conductor: \(7.42608\)
Root analytic conductor: \(2.72508\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{930} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 930,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.223345781\)
\(L(\frac12)\) \(\approx\) \(2.223345781\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
31 \( 1 + T \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 5 T + p T^{2} \)
23 \( 1 - 5 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 11 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 3 T + p T^{2} \)
79 \( 1 + 7 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.25069719725221377466344362644, −9.280020980784116097299413057805, −8.280444601616760167837315202360, −7.14865555445065532452689589221, −6.82214672330097626185245453025, −5.45443905087010954704044093941, −4.74130752956692692440196387757, −4.10337014083928661947731582719, −2.64510459623110775751468942168, −1.21930154759107999167274331762, 1.21930154759107999167274331762, 2.64510459623110775751468942168, 4.10337014083928661947731582719, 4.74130752956692692440196387757, 5.45443905087010954704044093941, 6.82214672330097626185245453025, 7.14865555445065532452689589221, 8.280444601616760167837315202360, 9.280020980784116097299413057805, 10.25069719725221377466344362644

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