Properties

Label 2-56350-1.1-c1-0-40
Degree $2$
Conductor $56350$
Sign $-1$
Analytic cond. $449.957$
Root an. cond. $21.2121$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 3·9-s − 2·11-s + 2·13-s + 16-s − 2·17-s − 3·18-s − 4·19-s − 2·22-s − 23-s + 2·26-s + 6·29-s − 2·31-s + 32-s − 2·34-s − 3·36-s − 4·37-s − 4·38-s + 12·41-s − 6·43-s − 2·44-s − 46-s + 6·47-s + 2·52-s + 12·53-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.353·8-s − 9-s − 0.603·11-s + 0.554·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s − 0.917·19-s − 0.426·22-s − 0.208·23-s + 0.392·26-s + 1.11·29-s − 0.359·31-s + 0.176·32-s − 0.342·34-s − 1/2·36-s − 0.657·37-s − 0.648·38-s + 1.87·41-s − 0.914·43-s − 0.301·44-s − 0.147·46-s + 0.875·47-s + 0.277·52-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56350\)    =    \(2 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(449.957\)
Root analytic conductor: \(21.2121\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 56350,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.65060631635100, −13.97881329534598, −13.71641910934768, −13.18349330198656, −12.63827633114610, −12.15408138861924, −11.63490308911354, −11.11872543839433, −10.52526641636605, −10.39813394289717, −9.423610132979513, −8.740988573046198, −8.531768408682384, −7.807474216309793, −7.266295788698300, −6.547373272256775, −6.070382581866028, −5.649305802131999, −4.990654346556279, −4.396753579590640, −3.825213417037990, −3.109811598431891, −2.522900701334638, −2.027020839973339, −0.9603923316295676, 0, 0.9603923316295676, 2.027020839973339, 2.522900701334638, 3.109811598431891, 3.825213417037990, 4.396753579590640, 4.990654346556279, 5.649305802131999, 6.070382581866028, 6.547373272256775, 7.266295788698300, 7.807474216309793, 8.531768408682384, 8.740988573046198, 9.423610132979513, 10.39813394289717, 10.52526641636605, 11.11872543839433, 11.63490308911354, 12.15408138861924, 12.63827633114610, 13.18349330198656, 13.71641910934768, 13.97881329534598, 14.65060631635100

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