Properties

Label 2-2645-1.1-c1-0-158
Degree $2$
Conductor $2645$
Sign $-1$
Analytic cond. $21.1204$
Root an. cond. $4.59569$
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  + 2·2-s + 2·4-s + 5-s − 7-s − 3·9-s + 2·10-s − 2·11-s − 2·13-s − 2·14-s − 4·16-s − 3·17-s − 6·18-s + 2·19-s + 2·20-s − 4·22-s + 25-s − 4·26-s − 2·28-s + 7·29-s − 5·31-s − 8·32-s − 6·34-s − 35-s − 6·36-s − 11·37-s + 4·38-s + 41-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s − 0.377·7-s − 9-s + 0.632·10-s − 0.603·11-s − 0.554·13-s − 0.534·14-s − 16-s − 0.727·17-s − 1.41·18-s + 0.458·19-s + 0.447·20-s − 0.852·22-s + 1/5·25-s − 0.784·26-s − 0.377·28-s + 1.29·29-s − 0.898·31-s − 1.41·32-s − 1.02·34-s − 0.169·35-s − 36-s − 1.80·37-s + 0.648·38-s + 0.156·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2645 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2645\)    =    \(5 \cdot 23^{2}\)
Sign: $-1$
Analytic conductor: \(21.1204\)
Root analytic conductor: \(4.59569\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2645,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
23 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
3 \( 1 + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 - T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 5 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−8.551506056517706907038082013704, −7.49497922965664040486613733245, −6.56389614092243760139893628224, −6.05404124546789425257449477618, −5.13478410924635941128044754757, −4.81791042909742149414815821233, −3.52621673706726526297889832836, −2.92211424763431944963205262512, −2.08330470038331107592069998888, 0, 2.08330470038331107592069998888, 2.92211424763431944963205262512, 3.52621673706726526297889832836, 4.81791042909742149414815821233, 5.13478410924635941128044754757, 6.05404124546789425257449477618, 6.56389614092243760139893628224, 7.49497922965664040486613733245, 8.551506056517706907038082013704

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