↑

Properties

Label 2-350-35.32-c2-0-9
Degree $2$
Conductor $350$
Sign $-0.993 + 0.114i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.366 + 1.36i)2-s + (1.06 + 3.96i)3-s + (−1.73 − i)4-s − 5.80·6-s + (6.83 + 1.48i)7-s + (2 − 1.99i)8-s + (−6.81 + 3.93i)9-s + (−5.51 + 9.54i)11-s + (2.12 − 7.93i)12-s + (−11.4 + 11.4i)13-s + (−4.53 + 8.79i)14-s + (1.99 + 3.46i)16-s + (15.7 − 4.22i)17-s + (−2.88 − 10.7i)18-s + (−23.3 + 13.5i)19-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (0.354 + 1.32i)3-s + (−0.433 − 0.250i)4-s − 0.968·6-s + (0.977 + 0.212i)7-s + (0.250 − 0.249i)8-s + (−0.757 + 0.437i)9-s + (−0.501 + 0.868i)11-s + (0.177 − 0.661i)12-s + (−0.877 + 0.877i)13-s + (−0.324 + 0.628i)14-s + (0.124 + 0.216i)16-s + (0.926 − 0.248i)17-s + (−0.160 − 0.597i)18-s + (−1.23 + 0.710i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.993 + 0.114i$
Analytic conductor: \(9.53680\)
Root analytic conductor: \(3.08817\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1),\ -0.993 + 0.114i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0827540 - 1.43976i\)
\(L(\frac12)\) \(\approx\) \(0.0827540 - 1.43976i\)
\(L(2)\) 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 \( 1 + (-6.83 - 1.48i)T \)
good3 \( 1 + (-1.06 - 3.96i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (5.51 - 9.54i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (11.4 - 11.4i)T - 169iT^{2} \)
17 \( 1 + (-15.7 + 4.22i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (23.3 - 13.5i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (20.9 + 5.61i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 13.2iT - 841T^{2} \)
31 \( 1 + (-24.6 + 42.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (4.12 - 15.3i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 + 23.0T + 1.68e3T^{2} \)
43 \( 1 + (-35.1 + 35.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-13.6 + 51.1i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-15.6 - 58.3i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (13.7 + 7.96i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.9 - 29.2i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-84.0 + 22.5i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 123.T + 5.04e3T^{2} \)
73 \( 1 + (-28.4 - 106. i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (109. - 63.2i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (6.82 - 6.82i)T - 6.88e3iT^{2} \)
89 \( 1 + (-49.9 + 28.8i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (-29.2 - 29.2i)T + 9.40e3iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.65824855654196747935681265278, −10.30328740408597371154482561880, −9.995848230729430061836616463915, −8.946789823625778376928725402915, −8.120586250874783081776536535911, −7.18410328966199329634667786594, −5.69177752758431921529032850940, −4.68767452690474800820462635947, −4.10244256028976271507117248016, −2.20000339595977187642254244524, 0.67424128065431592907823678080, 1.97126035102247110604004305139, 3.03509218132403290118748586557, 4.69162884391002624083456486589, 5.94186677194063472256454571645, 7.33200921182168878033811514112, 8.055839884574720188500691365665, 8.569737125125519151634636983560, 10.09510694685808621307282567336, 10.84569706366067070724012495296

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