Properties

Label 2-1-1.1-c21-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $2.79477$
Root an. cond. $1.67175$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 288·2-s − 1.28e5·3-s − 2.01e6·4-s + 2.16e7·5-s + 3.71e7·6-s − 7.68e8·7-s + 1.18e9·8-s + 6.14e9·9-s − 6.23e9·10-s − 9.47e10·11-s + 2.59e11·12-s − 8.06e10·13-s + 2.21e11·14-s − 2.78e12·15-s + 3.88e12·16-s + 3.05e12·17-s − 1.76e12·18-s − 7.92e12·19-s − 4.35e13·20-s + 9.89e13·21-s + 2.72e13·22-s − 7.38e13·23-s − 1.52e14·24-s − 8.50e12·25-s + 2.32e13·26-s + 5.56e14·27-s + 1.54e15·28-s + ⋯
L(s)  = 1  − 0.198·2-s − 1.25·3-s − 0.960·4-s + 0.991·5-s + 0.250·6-s − 1.02·7-s + 0.389·8-s + 0.587·9-s − 0.197·10-s − 1.10·11-s + 1.20·12-s − 0.162·13-s + 0.204·14-s − 1.24·15-s + 0.882·16-s + 0.367·17-s − 0.116·18-s − 0.296·19-s − 0.951·20-s + 1.29·21-s + 0.218·22-s − 0.371·23-s − 0.491·24-s − 0.0178·25-s + 0.0322·26-s + 0.520·27-s + 0.987·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(22-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+21/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(2.79477\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :21/2),\ -1)\)

Particular Values

\(L(11)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 9 p^{5} T + p^{21} T^{2} \)
3 \( 1 + 4772 p^{3} T + p^{21} T^{2} \)
5 \( 1 - 865638 p^{2} T + p^{21} T^{2} \)
7 \( 1 + 109725544 p T + p^{21} T^{2} \)
11 \( 1 + 94724929188 T + p^{21} T^{2} \)
13 \( 1 + 6201676138 p T + p^{21} T^{2} \)
17 \( 1 - 179546054706 p T + p^{21} T^{2} \)
19 \( 1 + 416883597460 p T + p^{21} T^{2} \)
23 \( 1 + 73845437470344 T + p^{21} T^{2} \)
29 \( 1 + 4253031736469010 T + p^{21} T^{2} \)
31 \( 1 - 1900541176310432 T + p^{21} T^{2} \)
37 \( 1 - 22191429912035222 T + p^{21} T^{2} \)
41 \( 1 + 20622803144546358 T + p^{21} T^{2} \)
43 \( 1 + 193605854685795844 T + p^{21} T^{2} \)
47 \( 1 - 146960504315611632 T + p^{21} T^{2} \)
53 \( 1 - 2038267110310687206 T + p^{21} T^{2} \)
59 \( 1 + 5975882742742352820 T + p^{21} T^{2} \)
61 \( 1 - 6190617154478149262 T + p^{21} T^{2} \)
67 \( 1 - 16961315295446680052 T + p^{21} T^{2} \)
71 \( 1 + 5632758963952293528 T + p^{21} T^{2} \)
73 \( 1 + 43284759511102937494 T + p^{21} T^{2} \)
79 \( 1 + 51264938664949064560 T + p^{21} T^{2} \)
83 \( 1 - 48911854702961049156 T + p^{21} T^{2} \)
89 \( 1 + \)\(50\!\cdots\!30\)\( T + p^{21} T^{2} \)
97 \( 1 - \)\(80\!\cdots\!82\)\( T + p^{21} T^{2} \)
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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

−28.21552381382824807188356139675, −26.04479803143780024778522315397, −23.23700442031546420878520342439, −21.92948421237360074119882630095, −18.35090669772360878294077021408, −16.86788900906693786397372094626, −13.09999180596557384531192800304, −10.00257819717286055683606436864, −5.62147606225101167443889903967, 0, 5.62147606225101167443889903967, 10.00257819717286055683606436864, 13.09999180596557384531192800304, 16.86788900906693786397372094626, 18.35090669772360878294077021408, 21.92948421237360074119882630095, 23.23700442031546420878520342439, 26.04479803143780024778522315397, 28.21552381382824807188356139675

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