Properties

Label 2-105-7.4-c3-0-4
Degree $2$
Conductor $105$
Sign $-0.653 - 0.757i$
Analytic cond. $6.19520$
Root an. cond. $2.48901$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 + 2.45i)2-s + (−1.5 + 2.59i)3-s + (−0.0184 + 0.0319i)4-s + (2.5 + 4.33i)5-s − 8.50·6-s + (4.61 + 17.9i)7-s + 22.5·8-s + (−4.5 − 7.79i)9-s + (−7.08 + 12.2i)10-s + (−31.3 + 54.3i)11-s + (−0.0554 − 0.0959i)12-s − 55.9·13-s + (−37.4 + 36.7i)14-s − 15.0·15-s + (32.1 + 55.6i)16-s + (63.4 − 109. i)17-s + ⋯
L(s)  = 1  + (0.501 + 0.868i)2-s + (−0.288 + 0.499i)3-s + (−0.00230 + 0.00399i)4-s + (0.223 + 0.387i)5-s − 0.578·6-s + (0.249 + 0.968i)7-s + 0.997·8-s + (−0.166 − 0.288i)9-s + (−0.224 + 0.388i)10-s + (−0.860 + 1.48i)11-s + (−0.00133 − 0.00230i)12-s − 1.19·13-s + (−0.715 + 0.701i)14-s − 0.258·15-s + (0.502 + 0.870i)16-s + (0.905 − 1.56i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $-0.653 - 0.757i$
Analytic conductor: \(6.19520\)
Root analytic conductor: \(2.48901\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{105} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :3/2),\ -0.653 - 0.757i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.809702 + 1.76833i\)
\(L(\frac12)\) \(\approx\) \(0.809702 + 1.76833i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 2.59i)T \)
5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 + (-4.61 - 17.9i)T \)
good2 \( 1 + (-1.41 - 2.45i)T + (-4 + 6.92i)T^{2} \)
11 \( 1 + (31.3 - 54.3i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 55.9T + 2.19e3T^{2} \)
17 \( 1 + (-63.4 + 109. i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-9.71 - 16.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-18.8 - 32.5i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 241.T + 2.43e4T^{2} \)
31 \( 1 + (33.2 - 57.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-24.7 - 42.9i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 4.73T + 6.89e4T^{2} \)
43 \( 1 - 460.T + 7.95e4T^{2} \)
47 \( 1 + (154. + 268. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-268. + 465. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-375. + 649. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (84.5 + 146. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-236. + 409. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 805.T + 3.57e5T^{2} \)
73 \( 1 + (127. - 220. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-208. - 360. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 108.T + 5.71e5T^{2} \)
89 \( 1 + (-248. - 430. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 1.52e3T + 9.12e5T^{2} \)
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

−14.13731965144658536316920024122, −12.62850370745221907921107812738, −11.70803863824962911884734218370, −10.25246322319200788028899613088, −9.612705317546744463638500465939, −7.75594721747347469448853785578, −6.83794392000222705251510821322, −5.31774172856040020800394195752, −4.89540705714652550659377813731, −2.47634049253834188835748739228, 1.02838299717990802885501872602, 2.80481376191898164010301756676, 4.35946362263168456374938764499, 5.74242343309382326188214269000, 7.41867290857320912022643391109, 8.285869063608823843949474222554, 10.27493717163845000416945977461, 10.83714129153947438869243391470, 12.02087089718551375889953101364, 12.83278759351129276782887294106

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