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Properties

Label 2-350-35.23-c2-0-17
Degree $2$
Conductor $350$
Sign $-0.996 + 0.0872i$
Analytic cond. $9.53680$
Root an. cond. $3.08817$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 − 1.36i)2-s + (−0.303 + 1.13i)3-s + (−1.73 + i)4-s + 1.65·6-s + (−4.91 − 4.97i)7-s + (2 + 1.99i)8-s + (6.60 + 3.81i)9-s + (−6.06 − 10.5i)11-s + (−0.606 − 2.26i)12-s + (8.70 + 8.70i)13-s + (−5.00 + 8.54i)14-s + (1.99 − 3.46i)16-s + (−5.40 − 1.44i)17-s + (2.79 − 10.4i)18-s + (−26.9 − 15.5i)19-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.101 + 0.376i)3-s + (−0.433 + 0.250i)4-s + 0.275·6-s + (−0.702 − 0.711i)7-s + (0.250 + 0.249i)8-s + (0.734 + 0.423i)9-s + (−0.551 − 0.955i)11-s + (−0.0505 − 0.188i)12-s + (0.669 + 0.669i)13-s + (−0.357 + 0.610i)14-s + (0.124 − 0.216i)16-s + (−0.318 − 0.0852i)17-s + (0.155 − 0.578i)18-s + (−1.41 − 0.817i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.996 + 0.0872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.996 + 0.0872i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.996 + 0.0872i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0203675 - 0.466094i\)
\(L(\frac12)\) \(\approx\) \(0.0203675 - 0.466094i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.366 + 1.36i)T \)
5 \( 1 \)
7 \( 1 + (4.91 + 4.97i)T \)
good3 \( 1 + (0.303 - 1.13i)T + (-7.79 - 4.5i)T^{2} \)
11 \( 1 + (6.06 + 10.5i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (-8.70 - 8.70i)T + 169iT^{2} \)
17 \( 1 + (5.40 + 1.44i)T + (250. + 144.5i)T^{2} \)
19 \( 1 + (26.9 + 15.5i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (30.1 - 8.07i)T + (458. - 264.5i)T^{2} \)
29 \( 1 + 25.2iT - 841T^{2} \)
31 \( 1 + (13.4 + 23.2i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + (14.8 + 55.5i)T + (-1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 45.8T + 1.68e3T^{2} \)
43 \( 1 + (18.2 + 18.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (-1.44 - 5.40i)T + (-1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (8.58 - 32.0i)T + (-2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (-51.0 + 29.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (37.7 - 65.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-54.1 - 14.5i)T + (3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 22.2T + 5.04e3T^{2} \)
73 \( 1 + (-0.862 + 3.21i)T + (-4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (37.4 + 21.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-35.1 - 35.1i)T + 6.88e3iT^{2} \)
89 \( 1 + (-11.8 - 6.85i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-58.3 + 58.3i)T - 9.40e3iT^{2} \)
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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

−10.76184264356573515897909936659, −10.10982129800597873286191956446, −9.168676786031983957550043318699, −8.190867530427875175887893837312, −7.03051251318319094553928334627, −5.88366788843067780795081995913, −4.37533011745857318653260985214, −3.69731706279691591945171962229, −2.10576688455264223440229481494, −0.22078123416638756730545541016, 1.86520926195608365142348070494, 3.65516123902749755983191842586, 5.00013295228814926704246218196, 6.23748154623291896374039325212, 6.74376104474842725103795531798, 7.981415013198990547785694418868, 8.723901157959917804148741896520, 9.945469322068349757212933672106, 10.40318015785200140730782051706, 12.02634762074942040549866662694

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