Properties

Label 2-64715-1.1-c1-0-1
Degree $2$
Conductor $64715$
Sign $1$
Analytic cond. $516.751$
Root an. cond. $22.7321$
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 + 7-s − 3·8-s − 2·9-s − 10-s − 12-s + 14-s − 15-s − 16-s − 2·17-s − 2·18-s − 4·19-s + 20-s + 21-s + 8·23-s − 3·24-s + 25-s − 5·27-s − 28-s + 3·29-s − 30-s + 10·31-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s − 2/3·9-s − 0.316·10-s − 0.288·12-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.471·18-s − 0.917·19-s + 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 0.962·27-s − 0.188·28-s + 0.557·29-s − 0.182·30-s + 1.79·31-s + 0.883·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64715 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(64715\)    =    \(5 \cdot 7 \cdot 43^{2}\)
Sign: $1$
Analytic conductor: \(516.751\)
Root analytic conductor: \(22.7321\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{64715} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 64715,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.159657996\)
\(L(\frac12)\) \(\approx\) \(2.159657996\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 - T \)
43 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−14.30109269778309, −13.61556099046888, −13.36773360724576, −12.90868744861101, −12.24743466565233, −11.80981409790777, −11.31664449705615, −10.81306983611795, −10.17033831758257, −9.460793214590584, −9.021642626325112, −8.458738741071171, −8.266717318750527, −7.635691207027407, −6.713617224028092, −6.442365095831044, −5.645694574887135, −5.104247263619718, −4.430327032510017, −4.272856906500729, −3.325254080719714, −2.899951214095906, −2.433410608856972, −1.328073502189051, −0.4408675677091872, 0.4408675677091872, 1.328073502189051, 2.433410608856972, 2.899951214095906, 3.325254080719714, 4.272856906500729, 4.430327032510017, 5.104247263619718, 5.645694574887135, 6.442365095831044, 6.713617224028092, 7.635691207027407, 8.266717318750527, 8.458738741071171, 9.021642626325112, 9.460793214590584, 10.17033831758257, 10.81306983611795, 11.31664449705615, 11.80981409790777, 12.24743466565233, 12.90868744861101, 13.36773360724576, 13.61556099046888, 14.30109269778309

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