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Properties

Label 2-153-153.106-c1-0-1
Degree $2$
Conductor $153$
Sign $0.691 - 0.721i$
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  + (−1.41 − 0.814i)2-s + (0.344 + 1.69i)3-s + (0.328 + 0.568i)4-s + (3.24 + 0.870i)5-s + (0.897 − 2.67i)6-s + (−4.04 + 1.08i)7-s + 2.18i·8-s + (−2.76 + 1.16i)9-s + (−3.87 − 3.87i)10-s + (3.68 − 0.988i)11-s + (−0.852 + 0.752i)12-s + (1.96 + 3.39i)13-s + (6.60 + 1.76i)14-s + (−0.359 + 5.81i)15-s + (2.44 − 4.22i)16-s + (1.76 + 3.72i)17-s + ⋯
L(s)  = 1  + (−0.998 − 0.576i)2-s + (0.198 + 0.980i)3-s + (0.164 + 0.284i)4-s + (1.45 + 0.389i)5-s + (0.366 − 1.09i)6-s + (−1.53 + 0.410i)7-s + 0.774i·8-s + (−0.921 + 0.389i)9-s + (−1.22 − 1.22i)10-s + (1.11 − 0.297i)11-s + (−0.246 + 0.217i)12-s + (0.544 + 0.942i)13-s + (1.76 + 0.472i)14-s + (−0.0928 + 1.50i)15-s + (0.610 − 1.05i)16-s + (0.427 + 0.903i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.691 - 0.721i$
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: \((2,\ 153,\ (\ :1/2),\ 0.691 - 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.707152 + 0.301756i\)
\(L(\frac12)\) \(\approx\) \(0.707152 + 0.301756i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.344 - 1.69i)T \)
17 \( 1 + (-1.76 - 3.72i)T \)
good2 \( 1 + (1.41 + 0.814i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-3.24 - 0.870i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (4.04 - 1.08i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (-3.68 + 0.988i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.96 - 3.39i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 0.510iT - 19T^{2} \)
23 \( 1 + (-0.309 + 1.15i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.735 + 2.74i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.60 + 1.23i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-4.73 + 4.73i)T - 37iT^{2} \)
41 \( 1 + (0.206 - 0.771i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.94 + 2.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.00 + 3.47i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 7.02iT - 53T^{2} \)
59 \( 1 + (-7.35 + 4.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.86 + 0.500i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (5.48 + 9.49i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.92 - 7.92i)T - 71iT^{2} \)
73 \( 1 + (7.02 - 7.02i)T - 73iT^{2} \)
79 \( 1 + (-5.60 + 1.50i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-11.6 - 6.74i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (0.387 + 1.44i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.20055421226066412757536275706, −11.69381412227753793886441382274, −10.64439580198914524103933655584, −9.826069665785314960335209832690, −9.351144124290480596682520570331, −8.713320175706866275647551256312, −6.43909274604704833718410316704, −5.70018782217748329035846375424, −3.60712682441561410186622812063, −2.16048679704408855882641505731, 1.12582050986989145367451558244, 3.25055342204962492338826216830, 5.88405026683343091169143741869, 6.57622789840884170837078833124, 7.49013540379581862476533040567, 8.937779082502662679875420555152, 9.423418710350320962825730372138, 10.28517882571135353150327926396, 12.17875232454320307270881840066, 13.17045941414421463761328225734

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