Properties

Label 2-1-1.1-c17-0-0
Degree $2$
Conductor $1$
Sign $-1$
Analytic cond. $1.83222$
Root an. cond. $1.35359$
Motivic weight $17$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 528·2-s − 4.28e3·3-s + 1.47e5·4-s − 1.02e6·5-s + 2.26e6·6-s + 3.22e6·7-s − 8.78e6·8-s − 1.10e8·9-s + 5.41e8·10-s − 7.53e8·11-s − 6.32e8·12-s + 2.54e9·13-s − 1.70e9·14-s + 4.39e9·15-s − 1.47e10·16-s − 5.42e9·17-s + 5.84e10·18-s + 1.48e9·19-s − 1.51e11·20-s − 1.38e10·21-s + 3.97e11·22-s − 3.17e11·23-s + 3.76e10·24-s + 2.89e11·25-s − 1.34e12·26-s + 1.02e12·27-s + 4.76e11·28-s + ⋯
L(s)  = 1  − 1.45·2-s − 0.376·3-s + 1.12·4-s − 1.17·5-s + 0.549·6-s + 0.211·7-s − 0.185·8-s − 0.857·9-s + 1.71·10-s − 1.06·11-s − 0.424·12-s + 0.863·13-s − 0.308·14-s + 0.442·15-s − 0.856·16-s − 0.188·17-s + 1.25·18-s + 0.0200·19-s − 1.32·20-s − 0.0797·21-s + 1.54·22-s − 0.844·23-s + 0.0697·24-s + 0.379·25-s − 1.26·26-s + 0.700·27-s + 0.238·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(18-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+17/2) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(1.83222\)
Root analytic conductor: \(1.35359\)
Motivic weight: \(17\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1,\ (\ :17/2),\ -1)\)

Particular Values

\(L(9)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 33 p^{4} T + p^{17} T^{2} \)
3 \( 1 + 476 p^{2} T + p^{17} T^{2} \)
5 \( 1 + 41034 p^{2} T + p^{17} T^{2} \)
7 \( 1 - 460856 p T + p^{17} T^{2} \)
11 \( 1 + 68510748 p T + p^{17} T^{2} \)
13 \( 1 - 195466502 p T + p^{17} T^{2} \)
17 \( 1 + 5429742318 T + p^{17} T^{2} \)
19 \( 1 - 1487499860 T + p^{17} T^{2} \)
23 \( 1 + 317091823464 T + p^{17} T^{2} \)
29 \( 1 - 2433410602590 T + p^{17} T^{2} \)
31 \( 1 + 8849722053088 T + p^{17} T^{2} \)
37 \( 1 - 12691652946662 T + p^{17} T^{2} \)
41 \( 1 - 48864151002282 T + p^{17} T^{2} \)
43 \( 1 + 91019974317844 T + p^{17} T^{2} \)
47 \( 1 + 49304994276048 T + p^{17} T^{2} \)
53 \( 1 - 22940453195766 T + p^{17} T^{2} \)
59 \( 1 - 32695090729980 T + p^{17} T^{2} \)
61 \( 1 + 1308285854869378 T + p^{17} T^{2} \)
67 \( 1 - 5196143861984132 T + p^{17} T^{2} \)
71 \( 1 + 3709489877412408 T + p^{17} T^{2} \)
73 \( 1 - 3402372968272586 T + p^{17} T^{2} \)
79 \( 1 - 2366533941308240 T + p^{17} T^{2} \)
83 \( 1 + 29766750443172204 T + p^{17} T^{2} \)
89 \( 1 - 29167184100574170 T + p^{17} T^{2} \)
97 \( 1 + 63769879140957598 T + p^{17} T^{2} \)
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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

−28.73387420325815890757109482027, −27.52981879333430235553227062819, −25.98046360545579795122352156785, −23.36658977788398006208160226429, −19.97810676051493623168994570962, −18.17341115038590061946085869072, −16.12038211220206065773617725346, −11.12333425499651648059374488485, −8.141610470203461898649318079038, 0, 8.141610470203461898649318079038, 11.12333425499651648059374488485, 16.12038211220206065773617725346, 18.17341115038590061946085869072, 19.97810676051493623168994570962, 23.36658977788398006208160226429, 25.98046360545579795122352156785, 27.52981879333430235553227062819, 28.73387420325815890757109482027

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