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Properties

Label 2-350-5.4-c3-0-20
Degree $2$
Conductor $350$
Sign $0.447 + 0.894i$
Analytic cond. $20.6506$
Root an. cond. $4.54430$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s − 3i·3-s − 4·4-s + 6·6-s + 7i·7-s − 8i·8-s + 18·9-s − 17·11-s + 12i·12-s − 81i·13-s − 14·14-s + 16·16-s + 91i·17-s + 36i·18-s − 102·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s + 0.377i·7-s − 0.353i·8-s + 0.666·9-s − 0.465·11-s + 0.288i·12-s − 1.72i·13-s − 0.267·14-s + 0.250·16-s + 1.29i·17-s + 0.471i·18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(20.6506\)
Root analytic conductor: \(4.54430\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.327899124\)
\(L(\frac12)\) \(\approx\) \(1.327899124\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 \)
7 \( 1 - 7iT \)
good3 \( 1 + 3iT - 27T^{2} \)
11 \( 1 + 17T + 1.33e3T^{2} \)
13 \( 1 + 81iT - 2.19e3T^{2} \)
17 \( 1 - 91iT - 4.91e3T^{2} \)
19 \( 1 + 102T + 6.85e3T^{2} \)
23 \( 1 + 90iT - 1.21e4T^{2} \)
29 \( 1 - 129T + 2.43e4T^{2} \)
31 \( 1 - 116T + 2.97e4T^{2} \)
37 \( 1 + 314iT - 5.06e4T^{2} \)
41 \( 1 + 124T + 6.89e4T^{2} \)
43 \( 1 + 434iT - 7.95e4T^{2} \)
47 \( 1 + 497iT - 1.03e5T^{2} \)
53 \( 1 + 584iT - 1.48e5T^{2} \)
59 \( 1 - 332T + 2.05e5T^{2} \)
61 \( 1 - 220T + 2.26e5T^{2} \)
67 \( 1 + 384iT - 3.00e5T^{2} \)
71 \( 1 + 664T + 3.57e5T^{2} \)
73 \( 1 - 230iT - 3.89e5T^{2} \)
79 \( 1 + 361T + 4.93e5T^{2} \)
83 \( 1 - 1.17e3iT - 5.71e5T^{2} \)
89 \( 1 + 40T + 7.04e5T^{2} \)
97 \( 1 - 175iT - 9.12e5T^{2} \)
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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

−10.53733944730544680693454600320, −10.17854673533669318148578442753, −8.512137212124069104945325526275, −8.177000964108885115914933420210, −7.02864123599620133905285995885, −6.15215326550742097832947971474, −5.18887093482956099563357642590, −3.86648157111727698568618872245, −2.23422264810574348477291435214, −0.49097821067059405007372558968, 1.41548116007541528333977614715, 2.87608480210429984746953956274, 4.32249910981479415825078058819, 4.71695331950196216249684466903, 6.42831728315554137605693528732, 7.46075813183918820099287549608, 8.758281515806361199995495573082, 9.616651139737712384209593572268, 10.23074031385991962403494144818, 11.23060983851067093203376671653

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