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Properties

Label 2-153-153.101-c2-0-7
Degree $2$
Conductor $153$
Sign $0.756 - 0.653i$
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  + (−0.726 − 0.419i)2-s + (2.58 + 1.52i)3-s + (−1.64 − 2.85i)4-s + (3.85 + 6.67i)5-s + (−1.23 − 2.19i)6-s + (−1.00 − 0.580i)7-s + 6.12i·8-s + (4.34 + 7.87i)9-s − 6.46i·10-s + (0.00243 − 0.00420i)11-s + (0.0954 − 9.88i)12-s + (3.65 + 6.32i)13-s + (0.487 + 0.843i)14-s + (−0.223 + 23.1i)15-s + (−4.02 + 6.96i)16-s + (10.4 − 13.3i)17-s + ⋯
L(s)  = 1  + (−0.363 − 0.209i)2-s + (0.861 + 0.508i)3-s + (−0.411 − 0.713i)4-s + (0.770 + 1.33i)5-s + (−0.206 − 0.365i)6-s + (−0.143 − 0.0829i)7-s + 0.765i·8-s + (0.483 + 0.875i)9-s − 0.646i·10-s + (0.000220 − 0.000382i)11-s + (0.00795 − 0.823i)12-s + (0.281 + 0.486i)13-s + (0.0347 + 0.0602i)14-s + (−0.0148 + 1.54i)15-s + (−0.251 + 0.435i)16-s + (0.616 − 0.787i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.756 - 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.756 - 0.653i$
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: \((2,\ 153,\ (\ :1),\ 0.756 - 0.653i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49820 + 0.557673i\)
\(L(\frac12)\) \(\approx\) \(1.49820 + 0.557673i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.58 - 1.52i)T \)
17 \( 1 + (-10.4 + 13.3i)T \)
good2 \( 1 + (0.726 + 0.419i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-3.85 - 6.67i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.00 + 0.580i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-0.00243 + 0.00420i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-3.65 - 6.32i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 - 4.00T + 361T^{2} \)
23 \( 1 + (-9.87 - 17.1i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (-16.7 + 28.9i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (12.9 - 7.50i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 58.7iT - 1.36e3T^{2} \)
41 \( 1 + (-3.62 - 6.28i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (15.8 - 27.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (59.1 + 34.1i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 55.8iT - 2.80e3T^{2} \)
59 \( 1 + (59.7 - 34.4i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (50.9 + 29.4i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (46.8 + 81.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 76.0T + 5.04e3T^{2} \)
73 \( 1 + 29.2iT - 5.32e3T^{2} \)
79 \( 1 + (-28.8 - 16.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-92.0 - 53.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 1.43iT - 7.92e3T^{2} \)
97 \( 1 + (-156. - 90.1i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−13.36377614172545260457340271757, −11.39157890256445568035965386466, −10.52668268994514410894248753873, −9.754863690738257713417062114234, −9.185421706191942515781741273567, −7.73045746024933978543471136846, −6.43449139650050833780963693375, −5.09447692056237680776814731795, −3.37030045700190771780352870822, −2.03035195737801788175579078956, 1.24011944183062908865477140630, 3.20791264943609523630321031133, 4.72893452390822687388562176027, 6.30183085480385694975198130829, 7.74720020466280784064744504520, 8.561086271448201063196716560218, 9.154318093136712968151090617926, 10.13292262170529218759020921100, 12.20170163197513127197839592595, 12.79721561368523030123866727422

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