↑

Properties

Label 2-350-35.3-c1-0-3
Degree $2$
Conductor $350$
Sign $0.626 - 0.779i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 + 0.258i)2-s + (0.0749 + 0.279i)3-s + (0.866 + 0.499i)4-s + 0.289i·6-s + (−0.126 + 2.64i)7-s + (0.707 + 0.707i)8-s + (2.52 − 1.45i)9-s + (−2.81 + 4.87i)11-s + (−0.0749 + 0.279i)12-s + (1.42 − 1.42i)13-s + (−0.806 + 2.51i)14-s + (0.500 + 0.866i)16-s + (5.12 − 1.37i)17-s + (2.81 − 0.754i)18-s + (−1.94 − 3.37i)19-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.0432 + 0.161i)3-s + (0.433 + 0.249i)4-s + 0.118i·6-s + (−0.0477 + 0.998i)7-s + (0.249 + 0.249i)8-s + (0.841 − 0.486i)9-s + (−0.848 + 1.46i)11-s + (−0.0216 + 0.0807i)12-s + (0.396 − 0.396i)13-s + (−0.215 + 0.673i)14-s + (0.125 + 0.216i)16-s + (1.24 − 0.333i)17-s + (0.663 − 0.177i)18-s + (−0.446 − 0.773i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 - 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.626 - 0.779i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83613 + 0.880571i\)
\(L(\frac12)\) \(\approx\) \(1.83613 + 0.880571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 \)
7 \( 1 + (0.126 - 2.64i)T \)
good3 \( 1 + (-0.0749 - 0.279i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (2.81 - 4.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.42 + 1.42i)T - 13iT^{2} \)
17 \( 1 + (-5.12 + 1.37i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.94 + 3.37i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.290 + 1.08i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 3.15iT - 29T^{2} \)
31 \( 1 + (3.33 + 1.92i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.86 + 1.30i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.21iT - 41T^{2} \)
43 \( 1 + (1.85 + 1.85i)T + 43iT^{2} \)
47 \( 1 + (-1.52 + 5.69i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.33 - 0.357i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.73 + 4.74i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.218 + 0.816i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.77T + 71T^{2} \)
73 \( 1 + (1.45 + 5.42i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.41 + 3.12i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.67 - 5.67i)T - 83iT^{2} \)
89 \( 1 + (5.96 + 10.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.63 - 6.63i)T + 97iT^{2} \)
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

−11.99590440915674589678278818019, −10.66706679019193153370058343772, −9.857028606280323672049256744549, −8.858576699072832162199391230905, −7.61499911390420569618092554118, −6.82783537727252845006394670135, −5.53822017560124765055766888762, −4.78884364798703974460790156727, −3.45586330988543940506173089988, −2.12140563053117587324289355439, 1.38699321160700900540241658913, 3.20793234310624120223114350449, 4.15637958391626481628815408827, 5.41239453496756022547601575966, 6.40806292605335077734280888462, 7.57379922323973288094017070500, 8.256622643382096537314474461626, 9.886988122205710352106758029468, 10.58275611248157217323502909766, 11.25833646977148951114504544241

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