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Properties

Label 2-153-153.101-c2-0-10
Degree $2$
Conductor $153$
Sign $-0.952 + 0.304i$
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  + (−3.33 − 1.92i)2-s + (−2.83 − 0.983i)3-s + (5.41 + 9.38i)4-s + (0.961 + 1.66i)5-s + (7.55 + 8.74i)6-s + (−1.81 − 1.04i)7-s − 26.3i·8-s + (7.06 + 5.57i)9-s − 7.40i·10-s + (8.01 − 13.8i)11-s + (−6.12 − 31.9i)12-s + (4.87 + 8.44i)13-s + (4.03 + 6.99i)14-s + (−1.08 − 5.66i)15-s + (−29.0 + 50.3i)16-s + (−16.9 + 1.77i)17-s + ⋯
L(s)  = 1  + (−1.66 − 0.962i)2-s + (−0.944 − 0.327i)3-s + (1.35 + 2.34i)4-s + (0.192 + 0.332i)5-s + (1.25 + 1.45i)6-s + (−0.259 − 0.149i)7-s − 3.29i·8-s + (0.784 + 0.619i)9-s − 0.740i·10-s + (0.728 − 1.26i)11-s + (−0.510 − 2.66i)12-s + (0.375 + 0.649i)13-s + (0.288 + 0.499i)14-s + (−0.0723 − 0.377i)15-s + (−1.81 + 3.14i)16-s + (−0.994 + 0.104i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $-0.952 + 0.304i$
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: \((2,\ 153,\ (\ :1),\ -0.952 + 0.304i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0477300 - 0.306416i\)
\(L(\frac12)\) \(\approx\) \(0.0477300 - 0.306416i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.83 + 0.983i)T \)
17 \( 1 + (16.9 - 1.77i)T \)
good2 \( 1 + (3.33 + 1.92i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-0.961 - 1.66i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (1.81 + 1.04i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-8.01 + 13.8i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-4.87 - 8.44i)T + (-84.5 + 146. i)T^{2} \)
19 \( 1 + 4.56T + 361T^{2} \)
23 \( 1 + (10.8 + 18.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 + (13.6 - 23.6i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (7.06 - 4.07i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 59.8iT - 1.36e3T^{2} \)
41 \( 1 + (18.6 + 32.2i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-31.9 + 55.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (49.9 + 28.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 63.4iT - 2.80e3T^{2} \)
59 \( 1 + (-9.56 + 5.51i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (71.3 + 41.1i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (15.9 + 27.5i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 108.T + 5.04e3T^{2} \)
73 \( 1 - 9.85iT - 5.32e3T^{2} \)
79 \( 1 + (-81.2 - 46.9i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (10.6 + 6.15i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 105. iT - 7.92e3T^{2} \)
97 \( 1 + (-106. - 61.7i)T + (4.70e3 + 8.14e3i)T^{2} \)
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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

−11.89089900297263506055281990869, −10.88843127820322260190629766286, −10.66403795939760029724494825229, −9.260913078227839520933229295606, −8.455415848004687532323848264485, −7.01834240771186155853939957001, −6.31955802688881840012631813560, −3.81166069190149712253460910761, −2.00774214173315169994967759429, −0.39875314144558595902501690230, 1.45587220703007701305989172287, 4.82567274957220468741773410915, 6.07906140888303200710983800141, 6.80761908124306417193771398736, 7.982783576425619907118161335854, 9.404321054863179442682515585383, 9.690501152491876270838819550806, 10.86849670846983087686371954254, 11.71544333782124880843096119154, 13.10451202067174761765408748650

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