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Properties

Label 2-2175-435.434-c0-0-11
Degree 22
Conductor 21752175
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 1.085461.08546
Root an. cond. 1.041851.04185
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.347i·2-s − i·3-s + 0.879·4-s + 0.347·6-s − 1.87i·7-s + 0.652i·8-s − 9-s + 1.53·11-s − 0.879i·12-s − 0.347i·13-s + 0.652·14-s + 0.652·16-s + 1.53i·17-s − 0.347i·18-s − 1.87·21-s + 0.532i·22-s + ⋯
L(s)  = 1  + 0.347i·2-s − i·3-s + 0.879·4-s + 0.347·6-s − 1.87i·7-s + 0.652i·8-s − 9-s + 1.53·11-s − 0.879i·12-s − 0.347i·13-s + 0.652·14-s + 0.652·16-s + 1.53i·17-s − 0.347i·18-s − 1.87·21-s + 0.532i·22-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 1.085461.08546
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2175(2174,⋅)\chi_{2175} (2174, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :0), 0.447+0.894i)(2,\ 2175,\ (\ :0),\ 0.447 + 0.894i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.5240969511.524096951
L(12)L(\frac12) ≈\approx 1.5240969511.524096951
L(1)L(1) 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+iT 1 + iT
5 1 1
29 1+T 1 + T
good2 1−0.347iT−T2 1 - 0.347iT - T^{2}
7 1+1.87iT−T2 1 + 1.87iT - T^{2}
11 1−1.53T+T2 1 - 1.53T + T^{2}
13 1+0.347iT−T2 1 + 0.347iT - T^{2}
17 1−1.53iT−T2 1 - 1.53iT - T^{2}
19 1−T2 1 - T^{2}
23 1+T2 1 + T^{2}
31 1−T2 1 - T^{2}
37 1+T2 1 + T^{2}
41 1+T+T2 1 + T + T^{2}
43 1+T2 1 + T^{2}
47 1+1.87iT−T2 1 + 1.87iT - T^{2}
53 1+T2 1 + T^{2}
59 1−T2 1 - T^{2}
61 1−T2 1 - T^{2}
67 1−1.53iT−T2 1 - 1.53iT - T^{2}
71 1−T2 1 - T^{2}
73 1+T2 1 + T^{2}
79 1−T2 1 - T^{2}
83 1+T2 1 + T^{2}
89 1+0.347T+T2 1 + 0.347T + T^{2}
97 1+T2 1 + 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

−8.818733295852830520035133972650, −8.061166441547019587320096293564, −7.41172586148492948770795137387, −6.76796647143613562922316230230, −6.40670099368601936077330933420, −5.44997141458835471154252212047, −3.99792092604343512027324767418, −3.43410170924670516091633517555, −1.88957024790543485481415868619, −1.17871366093912010891378725131, 1.78495838208498490317553901161, 2.74694215368942342595023696109, 3.41521212120751373256652689672, 4.54052864524554951160963920347, 5.47432548594210259895632437752, 6.14484266755393571404332150041, 6.86403047372750676836152242917, 8.017465875846241912170344301256, 9.136425061981662775916504624937, 9.219538216138578897846526317385

Graph of the ZZ-function along the critical line