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Properties

Label 2-2175-87.86-c0-0-8
Degree 22
Conductor 21752175
Sign −0.707+0.707i-0.707 + 0.707i
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  − 1.41·2-s + (−0.707 − 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (−0.707 − 0.707i)12-s − 0.999·16-s − 1.41·17-s − 1.41i·18-s + (0.707 − 0.707i)27-s − i·29-s + 1.41·32-s + 2.00·34-s + 1.00i·36-s − 1.41i·37-s + ⋯
L(s)  = 1  − 1.41·2-s + (−0.707 − 0.707i)3-s + 1.00·4-s + (1.00 + 1.00i)6-s + 1.00i·9-s + (−0.707 − 0.707i)12-s − 0.999·16-s − 1.41·17-s − 1.41i·18-s + (0.707 − 0.707i)27-s − i·29-s + 1.41·32-s + 2.00·34-s + 1.00i·36-s − 1.41i·37-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s) L(s)=((−0.707+0.707i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: −0.707+0.707i-0.707 + 0.707i
Analytic conductor: 1.085461.08546
Root analytic conductor: 1.041851.04185
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2175(1826,⋅)\chi_{2175} (1826, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2175, ( :0), −0.707+0.707i)(2,\ 2175,\ (\ :0),\ -0.707 + 0.707i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.24420977200.2442097720
L(12)L(\frac12) ≈\approx 0.24420977200.2442097720
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+(0.707+0.707i)T 1 + (0.707 + 0.707i)T
5 1 1
29 1+iT 1 + iT
good2 1+1.41T+T2 1 + 1.41T + T^{2}
7 1+T2 1 + T^{2}
11 1+T2 1 + T^{2}
13 1+T2 1 + T^{2}
17 1+1.41T+T2 1 + 1.41T + T^{2}
19 1−T2 1 - T^{2}
23 1−T2 1 - T^{2}
31 1−T2 1 - T^{2}
37 1+1.41iT−T2 1 + 1.41iT - T^{2}
41 1+T2 1 + T^{2}
43 1−1.41iT−T2 1 - 1.41iT - T^{2}
47 1−1.41T+T2 1 - 1.41T + T^{2}
53 1−T2 1 - T^{2}
59 1+2iT−T2 1 + 2iT - T^{2}
61 1−T2 1 - T^{2}
67 1+T2 1 + T^{2}
71 1+2iT−T2 1 + 2iT - T^{2}
73 1+1.41iT−T2 1 + 1.41iT - T^{2}
79 1−T2 1 - T^{2}
83 1−T2 1 - T^{2}
89 1+T2 1 + T^{2}
97 1−1.41iT−T2 1 - 1.41iT - 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

−9.042372603365892433783286145829, −8.032248482562974332471705275280, −7.68767066791053322335997440709, −6.71032443837427956329442847953, −6.26222535343257359988756493995, −5.08215947008945072946602121854, −4.20865028647409363998692113396, −2.48864770213640067956031190971, −1.67862085979285519302565276236, −0.33228161875003712152339817706, 1.22802583446277363449727970081, 2.59021122765571353649079088826, 3.97060540279298605551049685533, 4.73682795611349426892386075174, 5.68376103870545192860691440655, 6.74406962298863786701846641547, 7.14327671169323356247640484887, 8.388602377731503583936217095417, 8.841176608305671191361704665744, 9.511385970112094172872074819385

Graph of the ZZ-function along the critical line