↑

Properties

Label 2-2268-21.17-c1-0-0
Degree $2$
Conductor $2268$
Sign $-0.994 + 0.100i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
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  + (−0.0292 − 0.0505i)5-s + (−1.38 + 2.25i)7-s + (−0.578 − 0.333i)11-s + 2.60i·13-s + (3.62 − 6.28i)17-s + (−4.49 + 2.59i)19-s + (−4.90 + 2.83i)23-s + (2.49 − 4.32i)25-s + 9.41i·29-s + (−4.53 − 2.61i)31-s + (0.154 + 0.00395i)35-s + (−2.32 − 4.01i)37-s + 5.04·41-s − 5.66·43-s + (2.25 + 3.91i)47-s + ⋯
L(s)  = 1  + (−0.0130 − 0.0226i)5-s + (−0.521 + 0.852i)7-s + (−0.174 − 0.100i)11-s + 0.722i·13-s + (0.879 − 1.52i)17-s + (−1.03 + 0.595i)19-s + (−1.02 + 0.590i)23-s + (0.499 − 0.865i)25-s + 1.74i·29-s + (−0.814 − 0.470i)31-s + (0.0261 + 0.000668i)35-s + (−0.381 − 0.660i)37-s + 0.787·41-s − 0.864·43-s + (0.329 + 0.570i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.994 + 0.100i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (1781, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.994 + 0.100i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1769464077\)
\(L(\frac12)\) \(\approx\) \(0.1769464077\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (1.38 - 2.25i)T \)
good5 \( 1 + (0.0292 + 0.0505i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.578 + 0.333i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.60iT - 13T^{2} \)
17 \( 1 + (-3.62 + 6.28i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.49 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.90 - 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.41iT - 29T^{2} \)
31 \( 1 + (4.53 + 2.61i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.32 + 4.01i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 5.04T + 41T^{2} \)
43 \( 1 + 5.66T + 43T^{2} \)
47 \( 1 + (-2.25 - 3.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.72 + 3.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.45 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.86 - 5.11i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.26 - 3.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 + (3.61 + 2.08i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.68 + 8.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.84T + 83T^{2} \)
89 \( 1 + (4.29 + 7.44i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.33iT - 97T^{2} \)
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

−9.306924876294551881342147613045, −8.814375268498132016849299335981, −7.88217828365453305758153185314, −7.09298557857348400419068802283, −6.24539357969121591467143104031, −5.55047535104132629738779393226, −4.70458898413743554896682515631, −3.61687156383399410120983258692, −2.74538885552301387930165637809, −1.71729386416426199000648583897, 0.05850499546450157654624729779, 1.49232402499877525444609376785, 2.79178753513028591157362362719, 3.76631081430078157608793823794, 4.40700076394072503148670034130, 5.60644202985115418613581378904, 6.26672259514635601203712116404, 7.07636906714611188290118920093, 7.945538671312856510092087586887, 8.436024193769658737503304420287

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