Properties

Label 2-153-153.106-c1-0-12
Degree $2$
Conductor $153$
Sign $0.886 - 0.463i$
Analytic cond. $1.22171$
Root an. cond. $1.10531$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.07 + 1.19i)2-s + (0.641 − 1.60i)3-s + (1.85 + 3.22i)4-s + (−0.854 − 0.229i)5-s + (3.25 − 2.56i)6-s + (−2.38 + 0.639i)7-s + 4.11i·8-s + (−2.17 − 2.06i)9-s + (−1.49 − 1.49i)10-s + (−2.06 + 0.553i)11-s + (6.37 − 0.927i)12-s + (0.147 + 0.254i)13-s + (−5.70 − 1.52i)14-s + (−0.916 + 1.22i)15-s + (−1.19 + 2.07i)16-s + (3.94 + 1.20i)17-s + ⋯
L(s)  = 1  + (1.46 + 0.845i)2-s + (0.370 − 0.928i)3-s + (0.929 + 1.61i)4-s + (−0.382 − 0.102i)5-s + (1.32 − 1.04i)6-s + (−0.901 + 0.241i)7-s + 1.45i·8-s + (−0.726 − 0.687i)9-s + (−0.473 − 0.473i)10-s + (−0.622 + 0.166i)11-s + (1.84 − 0.267i)12-s + (0.0408 + 0.0707i)13-s + (−1.52 − 0.408i)14-s + (−0.236 + 0.317i)15-s + (−0.299 + 0.518i)16-s + (0.956 + 0.291i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.886 - 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.886 - 0.463i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 0.886 - 0.463i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.13130 + 0.523251i\)
\(L(\frac12)\) \(\approx\) \(2.13130 + 0.523251i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.641 + 1.60i)T \)
17 \( 1 + (-3.94 - 1.20i)T \)
good2 \( 1 + (-2.07 - 1.19i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (0.854 + 0.229i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (2.38 - 0.639i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.06 - 0.553i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.147 - 0.254i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 - 6.94iT - 19T^{2} \)
23 \( 1 + (-1.67 + 6.25i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.853 + 3.18i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-3.61 - 0.968i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.49 - 1.49i)T - 37iT^{2} \)
41 \( 1 + (-1.90 + 7.11i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-9.64 - 5.56i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.03 - 5.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.6iT - 53T^{2} \)
59 \( 1 + (5.41 - 3.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.22 + 2.47i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.110 + 0.191i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.53 - 7.53i)T - 71iT^{2} \)
73 \( 1 + (9.37 - 9.37i)T - 73iT^{2} \)
79 \( 1 + (10.8 - 2.90i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.97 - 1.71i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.17T + 89T^{2} \)
97 \( 1 + (-2.36 - 8.83i)T + (-84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.93499222444491827965374965048, −12.58940413253530779604546596621, −11.80925483503126687017758849999, −9.989398587320911086382668317855, −8.294053129439436941993826018806, −7.55477635150603785580452919513, −6.40066842275525374505666415201, −5.68858349184894621592229983968, −3.99292823583181451115149491053, −2.81370635501098067472873597245, 2.85586014435445456141530289522, 3.57462377452206606443476979870, 4.81232495722088252747328918068, 5.80931386084823509340126275450, 7.49111775124721895968914642109, 9.208173597611062599802374613582, 10.19977198123660905358732020550, 11.06611654316039695951957661052, 11.88002047000784506072538907221, 13.18040285425983878256805168289

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