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Properties

Label 2-2175-1.1-c3-0-258
Degree 22
Conductor 21752175
Sign 11
Analytic cond. 128.329128.329
Root an. cond. 11.328211.3282
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 22

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 3·3-s + 17·4-s − 15·6-s − 16·7-s − 45·8-s + 9·9-s − 44·11-s + 51·12-s − 78·13-s + 80·14-s + 89·16-s − 18·17-s − 45·18-s − 28·19-s − 48·21-s + 220·22-s − 184·23-s − 135·24-s + 390·26-s + 27·27-s − 272·28-s + 29·29-s − 224·31-s − 85·32-s − 132·33-s + 90·34-s + ⋯
L(s)  = 1  − 1.76·2-s + 0.577·3-s + 17/8·4-s − 1.02·6-s − 0.863·7-s − 1.98·8-s + 1/3·9-s − 1.20·11-s + 1.22·12-s − 1.66·13-s + 1.52·14-s + 1.39·16-s − 0.256·17-s − 0.589·18-s − 0.338·19-s − 0.498·21-s + 2.13·22-s − 1.66·23-s − 1.14·24-s + 2.94·26-s + 0.192·27-s − 1.83·28-s + 0.185·29-s − 1.29·31-s − 0.469·32-s − 0.696·33-s + 0.453·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 128.329128.329
Root analytic conductor: 11.328211.3282
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 22
Selberg data: (2, 2175, ( :3/2), 1)(2,\ 2175,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1−pT 1 - p T
5 1 1
29 1−pT 1 - p T
good2 1+5T+p3T2 1 + 5 T + p^{3} T^{2}
7 1+16T+p3T2 1 + 16 T + p^{3} T^{2}
11 1+4pT+p3T2 1 + 4 p T + p^{3} T^{2}
13 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
17 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
19 1+28T+p3T2 1 + 28 T + p^{3} T^{2}
23 1+8pT+p3T2 1 + 8 p T + p^{3} T^{2}
31 1+224T+p3T2 1 + 224 T + p^{3} T^{2}
37 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
41 1+78T+p3T2 1 + 78 T + p^{3} T^{2}
43 1−260T+p3T2 1 - 260 T + p^{3} T^{2}
47 1+312T+p3T2 1 + 312 T + p^{3} T^{2}
53 1+574T+p3T2 1 + 574 T + p^{3} T^{2}
59 1−180T+p3T2 1 - 180 T + p^{3} T^{2}
61 1+10pT+p3T2 1 + 10 p T + p^{3} T^{2}
67 1−340T+p3T2 1 - 340 T + p^{3} T^{2}
71 1−296T+p3T2 1 - 296 T + p^{3} T^{2}
73 1+394T+p3T2 1 + 394 T + p^{3} T^{2}
79 1+960T+p3T2 1 + 960 T + p^{3} T^{2}
83 1−908T+p3T2 1 - 908 T + p^{3} T^{2}
89 1+990T+p3T2 1 + 990 T + p^{3} T^{2}
97 1+1234T+p3T2 1 + 1234 T + p^{3} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−7.994998785593754847490276583999, −7.47924803281431699050672798748, −6.83391067655151459153220320592, −5.88951752091034193698067128248, −4.71678990498982780687333634282, −3.31739973970105445144356077472, −2.44968371113395567636761170728, −1.83528885658038055157024328758, 0, 0, 1.83528885658038055157024328758, 2.44968371113395567636761170728, 3.31739973970105445144356077472, 4.71678990498982780687333634282, 5.88951752091034193698067128248, 6.83391067655151459153220320592, 7.47924803281431699050672798748, 7.994998785593754847490276583999

Graph of the ZZ-function along the critical line