Properties

Label 2-770-1.1-c1-0-19
Degree $2$
Conductor $770$
Sign $-1$
Analytic cond. $6.14848$
Root an. cond. $2.47961$
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 − 2·3-s + 4-s − 5-s − 2·6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 2·12-s − 4·13-s + 14-s + 2·15-s + 16-s + 18-s − 4·19-s − 20-s − 2·21-s − 22-s − 2·24-s + 25-s − 4·26-s + 4·27-s + 28-s − 6·29-s + 2·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s + 0.516·15-s + 1/4·16-s + 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.436·21-s − 0.213·22-s − 0.408·24-s + 1/5·25-s − 0.784·26-s + 0.769·27-s + 0.188·28-s − 1.11·29-s + 0.365·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(770\)    =    \(2 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(6.14848\)
Root analytic conductor: \(2.47961\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{770} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 770,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
11 \( 1 + T \)
good3 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−10.31340102469523377302116267121, −9.080467819362277285025254767443, −7.88410539338216293675855384040, −7.11238985348297399935485729251, −6.19108363554867180855657086506, −5.24709944052927278525016542568, −4.72256680900554024987812021128, −3.52097826988955286053111224903, −2.04363202102792879733681514332, 0, 2.04363202102792879733681514332, 3.52097826988955286053111224903, 4.72256680900554024987812021128, 5.24709944052927278525016542568, 6.19108363554867180855657086506, 7.11238985348297399935485729251, 7.88410539338216293675855384040, 9.080467819362277285025254767443, 10.31340102469523377302116267121

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