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Properties

Label 2-153-153.101-c2-0-5
Degree $2$
Conductor $153$
Sign $0.676 - 0.736i$
Analytic cond. $4.16894$
Root an. cond. $2.04180$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.670 + 0.387i)2-s + (−2.92 + 0.669i)3-s + (−1.70 − 2.94i)4-s + (1.35 + 2.34i)5-s + (−2.22 − 0.683i)6-s + (8.30 + 4.79i)7-s − 5.73i·8-s + (8.10 − 3.91i)9-s + 2.09i·10-s + (−5.14 + 8.91i)11-s + (6.94 + 7.47i)12-s + (8.91 + 15.4i)13-s + (3.71 + 6.42i)14-s + (−5.52 − 5.94i)15-s + (−4.58 + 7.93i)16-s + (14.9 + 8.07i)17-s + ⋯
L(s)  = 1  + (0.335 + 0.193i)2-s + (−0.974 + 0.223i)3-s + (−0.425 − 0.736i)4-s + (0.270 + 0.468i)5-s + (−0.370 − 0.113i)6-s + (1.18 + 0.684i)7-s − 0.716i·8-s + (0.900 − 0.435i)9-s + 0.209i·10-s + (−0.467 + 0.810i)11-s + (0.578 + 0.622i)12-s + (0.685 + 1.18i)13-s + (0.265 + 0.459i)14-s + (−0.368 − 0.396i)15-s + (−0.286 + 0.496i)16-s + (0.880 + 0.474i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.676 - 0.736i$
Analytic conductor: \(4.16894\)
Root analytic conductor: \(2.04180\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1),\ 0.676 - 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.23432 + 0.542545i\)
\(L(\frac12)\) \(\approx\) \(1.23432 + 0.542545i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.92 - 0.669i)T \)
17 \( 1 + (-14.9 - 8.07i)T \)
good2 \( 1 + (-0.670 - 0.387i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-1.35 - 2.34i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-8.30 - 4.79i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (5.14 - 8.91i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-8.91 - 15.4i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 - 25.6T + 361T^{2} \)
23 \( 1 + (12.5 + 21.6i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (8.47 - 14.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (20.6 - 11.9i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 50.0iT - 1.36e3T^{2} \)
41 \( 1 + (-19.5 - 33.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-18.1 + 31.4i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (25.6 + 14.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 39.2iT - 2.80e3T^{2} \)
59 \( 1 + (64.6 - 37.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-23.7 - 13.6i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-53.3 - 92.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 25.8T + 5.04e3T^{2} \)
73 \( 1 + 58.8iT - 5.32e3T^{2} \)
79 \( 1 + (69.1 + 39.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (59.2 + 34.2i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 177. iT - 7.92e3T^{2} \)
97 \( 1 + (-24.6 - 14.2i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−12.75961265895118139402828498768, −11.82702242389232670637388434075, −10.84271876247035534681176308663, −10.04215899563511455277113624580, −8.951900838708494748029434695242, −7.29360803553376270844410119938, −6.06033956366271560583694593217, −5.25627996374975142030165744914, −4.27157893458210423570493316793, −1.61074423881299850812346217388, 1.06011330755127719121927550049, 3.51460923376713396706511219755, 5.03989491765640190126988916239, 5.57148049267996750570393279322, 7.63272482636016147949397137422, 8.044209292517870257647466804317, 9.688712497756335234122887454334, 11.04140264879480624627574684794, 11.52943896823717000592749650822, 12.66104967904275975237185666175

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