↑

Properties

Label 2-143-13.12-c3-0-17
Degree $2$
Conductor $143$
Sign $-0.920 - 0.391i$
Analytic cond. $8.43727$
Root an. cond. $2.90469$
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  + 4.83i·2-s + 9.74·3-s − 15.3·4-s + 8.44i·5-s + 47.0i·6-s + 12.5i·7-s − 35.5i·8-s + 67.9·9-s − 40.7·10-s − 11i·11-s − 149.·12-s + (18.3 − 43.1i)13-s − 60.7·14-s + 82.2i·15-s + 48.8·16-s − 47.4·17-s + ⋯
L(s)  = 1  + 1.70i·2-s + 1.87·3-s − 1.91·4-s + 0.755i·5-s + 3.20i·6-s + 0.679i·7-s − 1.57i·8-s + 2.51·9-s − 1.28·10-s − 0.301i·11-s − 3.59·12-s + (0.391 − 0.920i)13-s − 1.16·14-s + 1.41i·15-s + 0.763·16-s − 0.676·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(143\)    =    \(11 \cdot 13\)
Sign: $-0.920 - 0.391i$
Analytic conductor: \(8.43727\)
Root analytic conductor: \(2.90469\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{143} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 143,\ (\ :3/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.549240 + 2.69299i\)
\(L(\frac12)\) \(\approx\) \(0.549240 + 2.69299i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + 11iT \)
13 \( 1 + (-18.3 + 43.1i)T \)
good2 \( 1 - 4.83iT - 8T^{2} \)
3 \( 1 - 9.74T + 27T^{2} \)
5 \( 1 - 8.44iT - 125T^{2} \)
7 \( 1 - 12.5iT - 343T^{2} \)
17 \( 1 + 47.4T + 4.91e3T^{2} \)
19 \( 1 + 15.4iT - 6.85e3T^{2} \)
23 \( 1 + 6.78T + 1.21e4T^{2} \)
29 \( 1 + 263.T + 2.43e4T^{2} \)
31 \( 1 - 140. iT - 2.97e4T^{2} \)
37 \( 1 + 389. iT - 5.06e4T^{2} \)
41 \( 1 - 235. iT - 6.89e4T^{2} \)
43 \( 1 - 523.T + 7.95e4T^{2} \)
47 \( 1 + 247. iT - 1.03e5T^{2} \)
53 \( 1 - 588.T + 1.48e5T^{2} \)
59 \( 1 - 464. iT - 2.05e5T^{2} \)
61 \( 1 + 266.T + 2.26e5T^{2} \)
67 \( 1 - 76.0iT - 3.00e5T^{2} \)
71 \( 1 + 497. iT - 3.57e5T^{2} \)
73 \( 1 + 422. iT - 3.89e5T^{2} \)
79 \( 1 + 1.18e3T + 4.93e5T^{2} \)
83 \( 1 - 380. iT - 5.71e5T^{2} \)
89 \( 1 - 744. iT - 7.04e5T^{2} \)
97 \( 1 + 532. iT - 9.12e5T^{2} \)
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

−13.57369437710962714980790356199, −12.81840371292509812041331096329, −10.66320587351101418841710595552, −9.204979173809099460895263195318, −8.744679738140785376068672946497, −7.72686013204227113676483986333, −7.01606083898674204855841090995, −5.66805143290194598850496818896, −3.95829840820640547691024698438, −2.61356735533434591662848393908, 1.32653102961506513928609573533, 2.40086107444904791328613355698, 3.82639909598437019334195328060, 4.40800987169971668590993899037, 7.28785113030059956032812373635, 8.598009578063975008280776823236, 9.198616948735835508160570000050, 9.970147422468245983812280157402, 11.11913951366141014854464136528, 12.43206272152667186861020911429

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