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Properties

Label 2-153-153.106-c1-0-7
Degree $2$
Conductor $153$
Sign $0.396 - 0.917i$
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  + (1.25 + 0.727i)2-s + (0.0948 + 1.72i)3-s + (0.0574 + 0.0995i)4-s + (1.43 + 0.384i)5-s + (−1.13 + 2.24i)6-s + (1.42 − 0.382i)7-s − 2.74i·8-s + (−2.98 + 0.328i)9-s + (1.52 + 1.52i)10-s + (−5.85 + 1.56i)11-s + (−0.166 + 0.108i)12-s + (0.466 + 0.807i)13-s + (2.07 + 0.556i)14-s + (−0.529 + 2.52i)15-s + (2.10 − 3.65i)16-s + (3.65 + 1.91i)17-s + ⋯
L(s)  = 1  + (0.890 + 0.514i)2-s + (0.0547 + 0.998i)3-s + (0.0287 + 0.0497i)4-s + (0.642 + 0.172i)5-s + (−0.464 + 0.917i)6-s + (0.540 − 0.144i)7-s − 0.969i·8-s + (−0.994 + 0.109i)9-s + (0.483 + 0.483i)10-s + (−1.76 + 0.473i)11-s + (−0.0481 + 0.0314i)12-s + (0.129 + 0.224i)13-s + (0.555 + 0.148i)14-s + (−0.136 + 0.650i)15-s + (0.527 − 0.912i)16-s + (0.885 + 0.463i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.396 - 0.917i$
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: \((2,\ 153,\ (\ :1/2),\ 0.396 - 0.917i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44847 + 0.951758i\)
\(L(\frac12)\) \(\approx\) \(1.44847 + 0.951758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.0948 - 1.72i)T \)
17 \( 1 + (-3.65 - 1.91i)T \)
good2 \( 1 + (-1.25 - 0.727i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.43 - 0.384i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.42 + 0.382i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (5.85 - 1.56i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.466 - 0.807i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 + 6.88iT - 19T^{2} \)
23 \( 1 + (-1.42 + 5.30i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-2.23 - 8.33i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-1.47 - 0.394i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.700 + 0.700i)T - 37iT^{2} \)
41 \( 1 + (0.939 - 3.50i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (-0.360 - 0.208i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.62 - 6.28i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.69iT - 53T^{2} \)
59 \( 1 + (6.12 - 3.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.93 - 1.32i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.134 + 0.232i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.80 + 2.80i)T - 71iT^{2} \)
73 \( 1 + (-9.68 + 9.68i)T - 73iT^{2} \)
79 \( 1 + (-4.79 + 1.28i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (0.784 + 0.452i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 0.553T + 89T^{2} \)
97 \( 1 + (-3.75 - 14.0i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.45561941586775556539967923612, −12.49314951633187076295237013455, −10.83186132536173239489928468003, −10.28826994811875251544459969656, −9.229058124702308277503239484860, −7.83951461274888822885100858300, −6.35604286585668804975245203485, −5.15371728725854065387213326906, −4.65817467792400312217032219887, −2.90517777656255147038820595834, 2.01077005271729714740373605790, 3.26947660290853450395490265364, 5.28968288010489146764955302890, 5.77600981023168729358158160212, 7.82246111445845613618882350692, 8.194900596261601985418121613226, 9.936212594869442507059835985595, 11.21597810624708285528842473984, 12.04312173128120413211119916007, 12.92423642044410480531713013464

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