Properties

Label 2-3-1.1-c21-0-1
Degree $2$
Conductor $3$
Sign $-1$
Analytic cond. $8.38432$
Root an. cond. $2.89556$
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  − 2.84e3·2-s − 5.90e4·3-s + 5.99e6·4-s + 3.10e6·5-s + 1.67e8·6-s + 3.63e8·7-s − 1.10e10·8-s + 3.48e9·9-s − 8.84e9·10-s + 1.45e10·11-s − 3.53e11·12-s + 1.13e11·13-s − 1.03e12·14-s − 1.83e11·15-s + 1.89e13·16-s − 8.58e12·17-s − 9.91e12·18-s − 2.92e13·19-s + 1.86e13·20-s − 2.14e13·21-s − 4.14e13·22-s − 1.55e14·23-s + 6.53e14·24-s − 4.67e14·25-s − 3.22e14·26-s − 2.05e14·27-s + 2.17e15·28-s + ⋯
L(s)  = 1  − 1.96·2-s − 0.577·3-s + 2.85·4-s + 0.142·5-s + 1.13·6-s + 0.486·7-s − 3.64·8-s + 1/3·9-s − 0.279·10-s + 0.169·11-s − 1.64·12-s + 0.228·13-s − 0.954·14-s − 0.0822·15-s + 4.30·16-s − 1.03·17-s − 0.654·18-s − 1.09·19-s + 0.406·20-s − 0.280·21-s − 0.332·22-s − 0.784·23-s + 2.10·24-s − 0.979·25-s − 0.447·26-s − 0.192·27-s + 1.38·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-1$
Analytic conductor: \(8.38432\)
Root analytic conductor: \(2.89556\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3,\ (\ :21/2),\ -1)\)

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)$
bad3 \( 1 + p^{10} T \)
good2 \( 1 + 711 p^{2} T + p^{21} T^{2} \)
5 \( 1 - 124398 p^{2} T + p^{21} T^{2} \)
7 \( 1 - 51900560 p T + p^{21} T^{2} \)
11 \( 1 - 1325621196 p T + p^{21} T^{2} \)
13 \( 1 - 113350790702 T + p^{21} T^{2} \)
17 \( 1 + 505258211646 p T + p^{21} T^{2} \)
19 \( 1 + 1536996803884 p T + p^{21} T^{2} \)
23 \( 1 + 155899214954280 T + p^{21} T^{2} \)
29 \( 1 - 2400788707090758 T + p^{21} T^{2} \)
31 \( 1 - 2239820676947000 T + p^{21} T^{2} \)
37 \( 1 + 30785069383298890 T + p^{21} T^{2} \)
41 \( 1 + 103207571041281030 T + p^{21} T^{2} \)
43 \( 1 + 165557270617488124 T + p^{21} T^{2} \)
47 \( 1 + 66587216226477408 T + p^{21} T^{2} \)
53 \( 1 - 435422766592881630 T + p^{21} T^{2} \)
59 \( 1 - 5534365798259081316 T + p^{21} T^{2} \)
61 \( 1 + 7176205164722961202 T + p^{21} T^{2} \)
67 \( 1 + 15755449453068299812 T + p^{21} T^{2} \)
71 \( 1 - 26457854874259376232 T + p^{21} T^{2} \)
73 \( 1 - 13471249335464801450 T + p^{21} T^{2} \)
79 \( 1 + 16886125085525986840 T + p^{21} T^{2} \)
83 \( 1 + \)\(17\!\cdots\!72\)\( T + p^{21} T^{2} \)
89 \( 1 + \)\(31\!\cdots\!86\)\( T + p^{21} T^{2} \)
97 \( 1 - \)\(94\!\cdots\!18\)\( T + p^{21} T^{2} \)
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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

−19.64882535054477525047031686216, −18.10129991399174111376656103593, −17.12657950832048419183062978110, −15.57887440143012401593974921755, −11.67533953502725227132355006764, −10.27577724529991206035988500531, −8.434446208305040756622708334893, −6.52657195071322903272764533362, −1.80521254452922275171971322257, 0, 1.80521254452922275171971322257, 6.52657195071322903272764533362, 8.434446208305040756622708334893, 10.27577724529991206035988500531, 11.67533953502725227132355006764, 15.57887440143012401593974921755, 17.12657950832048419183062978110, 18.10129991399174111376656103593, 19.64882535054477525047031686216

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