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Properties

Label 2-350-5.4-c5-0-18
Degree 22
Conductor 350350
Sign 0.447+0.894i0.447 + 0.894i
Analytic cond. 56.134356.1343
Root an. cond. 7.492287.49228
Motivic weight 55
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 17i·3-s − 16·4-s − 68·6-s + 49i·7-s + 64i·8-s − 46·9-s − 715·11-s + 272i·12-s + 331i·13-s + 196·14-s + 256·16-s + 1.69e3i·17-s + 184i·18-s + 1.71e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.09i·3-s − 0.5·4-s − 0.771·6-s + 0.377i·7-s + 0.353i·8-s − 0.189·9-s − 1.78·11-s + 0.545i·12-s + 0.543i·13-s + 0.267·14-s + 0.250·16-s + 1.42i·17-s + 0.133i·18-s + 1.09·19-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=((0.447+0.894i) Λ‾(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+5/2) L(s)=((0.447+0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 0.447+0.894i0.447 + 0.894i
Analytic conductor: 56.134356.1343
Root analytic conductor: 7.492287.49228
Motivic weight: 55
Rational: no
Arithmetic: yes
Character: χ350(99,⋅)\chi_{350} (99, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 350, ( :5/2), 0.447+0.894i)(2,\ 350,\ (\ :5/2),\ 0.447 + 0.894i)

Particular Values

L(3)L(3) ≈\approx 1.5883863811.588386381
L(12)L(\frac12) ≈\approx 1.5883863811.588386381
L(72)L(\frac{7}{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)
bad2 1+4iT 1 + 4iT
5 1 1
7 1−49iT 1 - 49iT
good3 1+17iT−243T2 1 + 17iT - 243T^{2}
11 1+715T+1.61e5T2 1 + 715T + 1.61e5T^{2}
13 1−331iT−3.71e5T2 1 - 331iT - 3.71e5T^{2}
17 1−1.69e3iT−1.41e6T2 1 - 1.69e3iT - 1.41e6T^{2}
19 1−1.71e3T+2.47e6T2 1 - 1.71e3T + 2.47e6T^{2}
23 1+3.95e3iT−6.43e6T2 1 + 3.95e3iT - 6.43e6T^{2}
29 1+4.57e3T+2.05e7T2 1 + 4.57e3T + 2.05e7T^{2}
31 1−6.75e3T+2.86e7T2 1 - 6.75e3T + 2.86e7T^{2}
37 1−1.65e4iT−6.93e7T2 1 - 1.65e4iT - 6.93e7T^{2}
41 1−1.88e4T+1.15e8T2 1 - 1.88e4T + 1.15e8T^{2}
43 1−2.25e3iT−1.47e8T2 1 - 2.25e3iT - 1.47e8T^{2}
47 1−537iT−2.29e8T2 1 - 537iT - 2.29e8T^{2}
53 1+1.09e4iT−4.18e8T2 1 + 1.09e4iT - 4.18e8T^{2}
59 1−2.59e4T+7.14e8T2 1 - 2.59e4T + 7.14e8T^{2}
61 1−3.91e4T+8.44e8T2 1 - 3.91e4T + 8.44e8T^{2}
67 1+4.41e3iT−1.35e9T2 1 + 4.41e3iT - 1.35e9T^{2}
71 1+3.18e4T+1.80e9T2 1 + 3.18e4T + 1.80e9T^{2}
73 1+5.01e3iT−2.07e9T2 1 + 5.01e3iT - 2.07e9T^{2}
79 1−2.79e4T+3.07e9T2 1 - 2.79e4T + 3.07e9T^{2}
83 1−3.76e4iT−3.93e9T2 1 - 3.76e4iT - 3.93e9T^{2}
89 1−1.72e4T+5.58e9T2 1 - 1.72e4T + 5.58e9T^{2}
97 1−6.31e4iT−8.58e9T2 1 - 6.31e4iT - 8.58e9T^{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

−10.54164765101383559841887780534, −9.789823787617255118432247411140, −8.381581213438818775333932740666, −7.903656042881116029370356597644, −6.68627590212716470361602700298, −5.62237699697275029917362230310, −4.43241809883464917781612230318, −2.85038191261791535347148887050, −2.01283574758085709764277292845, −0.800794083188094380774345139492, 0.58102727409788131478990997147, 2.81557602060556826075794594913, 3.94805097309223833837832440191, 5.17148082133071264803394222048, 5.51441757194966960079643938234, 7.39442305784458136179207890557, 7.67772656745840568474321369764, 9.176747166514246907711093275198, 9.799375259608524416022015197766, 10.57752443192637267844597930175

Graph of the ZZ-function along the critical line