Properties

Label 2-60984-1.1-c1-0-43
Degree $2$
Conductor $60984$
Sign $-1$
Analytic cond. $486.959$
Root an. cond. $22.0671$
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  + 5-s − 7-s − 2·17-s + 2·19-s + 7·23-s − 4·25-s − 10·29-s + 7·31-s − 35-s − 9·37-s − 2·41-s + 4·43-s − 8·47-s + 49-s − 2·53-s + 15·59-s + 14·61-s + 3·67-s − 3·71-s − 10·73-s − 10·79-s − 2·85-s + 11·89-s + 2·95-s + 7·97-s + 101-s + 103-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.485·17-s + 0.458·19-s + 1.45·23-s − 4/5·25-s − 1.85·29-s + 1.25·31-s − 0.169·35-s − 1.47·37-s − 0.312·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s − 0.274·53-s + 1.95·59-s + 1.79·61-s + 0.366·67-s − 0.356·71-s − 1.17·73-s − 1.12·79-s − 0.216·85-s + 1.16·89-s + 0.205·95-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60984\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(486.959\)
Root analytic conductor: \(22.0671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{60984} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 60984,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 - T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 3 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 11 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−14.65884701787377, −13.97958527862793, −13.39768046511305, −13.10820434122518, −12.74785215618790, −11.89374981947003, −11.50997820626868, −11.10294121465128, −10.30804066919671, −9.997862488261035, −9.444348051979910, −8.899584715867101, −8.506378295084898, −7.726043838658613, −7.177280763358931, −6.724395708285619, −6.165725165434914, −5.391914294623577, −5.203034675709805, −4.325943120442058, −3.672106236064741, −3.140521972138192, −2.378557664732667, −1.781052101195630, −0.9467989290149556, 0, 0.9467989290149556, 1.781052101195630, 2.378557664732667, 3.140521972138192, 3.672106236064741, 4.325943120442058, 5.203034675709805, 5.391914294623577, 6.165725165434914, 6.724395708285619, 7.177280763358931, 7.726043838658613, 8.506378295084898, 8.899584715867101, 9.444348051979910, 9.997862488261035, 10.30804066919671, 11.10294121465128, 11.50997820626868, 11.89374981947003, 12.74785215618790, 13.10820434122518, 13.39768046511305, 13.97958527862793, 14.65884701787377

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