↑

Properties

Label 2-1260-105.2-c1-0-3
Degree $2$
Conductor $1260$
Sign $-0.430 - 0.902i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  + (−1.36 − 1.77i)5-s + (−1.95 − 1.77i)7-s + (−0.278 + 0.160i)11-s + (−3.06 + 3.06i)13-s + (7.47 + 2.00i)17-s + (−6.65 − 3.84i)19-s + (−0.170 + 0.0455i)23-s + (−1.26 + 4.83i)25-s − 3.35·29-s + (3.49 + 6.06i)31-s + (−0.470 + 5.89i)35-s + (−6.27 + 1.68i)37-s + 9.24i·41-s + (3.35 − 3.35i)43-s + (1.48 + 5.52i)47-s + ⋯
L(s)  = 1  + (−0.611 − 0.791i)5-s + (−0.740 − 0.672i)7-s + (−0.0840 + 0.0485i)11-s + (−0.850 + 0.850i)13-s + (1.81 + 0.486i)17-s + (−1.52 − 0.881i)19-s + (−0.0354 + 0.00950i)23-s + (−0.253 + 0.967i)25-s − 0.622·29-s + (0.628 + 1.08i)31-s + (−0.0796 + 0.996i)35-s + (−1.03 + 0.276i)37-s + 1.44i·41-s + (0.511 − 0.511i)43-s + (0.216 + 0.806i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.430 - 0.902i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.430 - 0.902i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3431556975\)
\(L(\frac12)\) \(\approx\) \(0.3431556975\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (1.36 + 1.77i)T \)
7 \( 1 + (1.95 + 1.77i)T \)
good11 \( 1 + (0.278 - 0.160i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.06 - 3.06i)T - 13iT^{2} \)
17 \( 1 + (-7.47 - 2.00i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (6.65 + 3.84i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.170 - 0.0455i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3.35T + 29T^{2} \)
31 \( 1 + (-3.49 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.27 - 1.68i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.24iT - 41T^{2} \)
43 \( 1 + (-3.35 + 3.35i)T - 43iT^{2} \)
47 \( 1 + (-1.48 - 5.52i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.18 + 8.16i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.20 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.19 - 7.27i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.54 - 5.76i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.74iT - 71T^{2} \)
73 \( 1 + (14.4 + 3.86i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.747 + 0.431i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.4 + 10.4i)T + 83iT^{2} \)
89 \( 1 + (3.91 - 6.77i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.08 - 5.08i)T + 97iT^{2} \)
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

−9.974495152623503038618035253685, −9.099123439378848963565106354187, −8.376792583999069957204539020392, −7.43606218459703178757517231110, −6.83340766608331415233211276643, −5.73449042173731277188915176795, −4.66549279515848740957549129764, −4.01647314263025867625124822315, −2.93778974362590726405539968666, −1.34026304183091010610213205131, 0.14832063780477487645467067687, 2.30760888912263035416860891124, 3.17798417595301841563881614320, 3.99940897062958687222081509359, 5.41030118656671357180501585561, 6.02019162308335257826635245407, 7.05721757851521913397976869916, 7.76550009170356229841969350083, 8.474318202502268024477438075147, 9.641255749385113071920531480485

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