↑

Properties

Label 2-252-252.103-c1-0-18
Degree $2$
Conductor $252$
Sign $-0.425 - 0.904i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (0.514 + 1.31i)2-s + (1.01 + 1.40i)3-s + (−1.47 + 1.35i)4-s + (2.53 − 1.46i)5-s + (−1.32 + 2.05i)6-s + (−2.33 + 1.24i)7-s + (−2.54 − 1.23i)8-s + (−0.943 + 2.84i)9-s + (3.23 + 2.59i)10-s + (1.58 + 0.915i)11-s + (−3.39 − 0.690i)12-s + (0.488 + 0.281i)13-s + (−2.84 − 2.43i)14-s + (4.63 + 2.07i)15-s + (0.324 − 3.98i)16-s + (0.330 − 0.191i)17-s + ⋯
L(s)  = 1  + (0.363 + 0.931i)2-s + (0.585 + 0.810i)3-s + (−0.735 + 0.677i)4-s + (1.13 − 0.655i)5-s + (−0.542 + 0.840i)6-s + (−0.881 + 0.471i)7-s + (−0.898 − 0.438i)8-s + (−0.314 + 0.949i)9-s + (1.02 + 0.819i)10-s + (0.478 + 0.275i)11-s + (−0.979 − 0.199i)12-s + (0.135 + 0.0782i)13-s + (−0.759 − 0.649i)14-s + (1.19 + 0.536i)15-s + (0.0811 − 0.996i)16-s + (0.0802 − 0.0463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $-0.425 - 0.904i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ -0.425 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950760 + 1.49806i\)
\(L(\frac12)\) \(\approx\) \(0.950760 + 1.49806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.514 - 1.31i)T \)
3 \( 1 + (-1.01 - 1.40i)T \)
7 \( 1 + (2.33 - 1.24i)T \)
good5 \( 1 + (-2.53 + 1.46i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.58 - 0.915i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.488 - 0.281i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.330 + 0.191i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.34 + 7.52i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.16 - 2.40i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.82 + 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 0.654T + 31T^{2} \)
37 \( 1 + (-3.63 + 6.30i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-8.94 - 5.16i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.57 + 3.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.97T + 47T^{2} \)
53 \( 1 + (3.65 + 6.32i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 0.0693T + 59T^{2} \)
61 \( 1 - 9.83iT - 61T^{2} \)
67 \( 1 + 1.81iT - 67T^{2} \)
71 \( 1 - 9.64iT - 71T^{2} \)
73 \( 1 + (13.1 - 7.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 6.31iT - 79T^{2} \)
83 \( 1 + (-2.90 - 5.02i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.91 + 1.10i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.59 + 0.918i)T + (48.5 - 84.0i)T^{2} \)
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

−12.84829917395256983124174092398, −11.50988563883694929528131678424, −9.696339770073124846224115928444, −9.485239130179774263084666763057, −8.720238516888075421668780219027, −7.37105550592351289297725878549, −6.04369621893828061798197720059, −5.29165165330993900389315583917, −4.11951431704940514335256388183, −2.70101258130713319384426704686, 1.48514535725414617025782104884, 2.83289217123441171358765735536, 3.76648789216772304567472047723, 5.89459419935525027734991263215, 6.37759153715438090081447801487, 7.83506904203832994248353817569, 9.262577832998828746411108647853, 9.833608942095138106981825880391, 10.70063964877832105881452358299, 11.99001587396583533543354126075

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