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Properties

Label 2-153-17.16-c1-0-2
Degree $2$
Conductor $153$
Sign $0.242 + 0.970i$
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  − 2-s − 4-s − 4i·7-s + 3·8-s − 4i·11-s + 2·13-s + 4i·14-s − 16-s + (−1 − 4i)17-s − 4·19-s + 4i·22-s + 4i·23-s + 5·25-s − 2·26-s + 4i·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.5·4-s − 1.51i·7-s + 1.06·8-s − 1.20i·11-s + 0.554·13-s + 1.06i·14-s − 0.250·16-s + (−0.242 − 0.970i)17-s − 0.917·19-s + 0.852i·22-s + 0.834i·23-s + 25-s − 0.392·26-s + 0.755i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(1.22171\)
Root analytic conductor: \(1.10531\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{153} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 153,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.507174 - 0.395989i\)
\(L(\frac12)\) \(\approx\) \(0.507174 - 0.395989i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 + (1 + 4i)T \)
good2 \( 1 + T + 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 4iT - 7T^{2} \)
11 \( 1 + 4iT - 11T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 - 4iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 8iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 4iT - 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 16iT - 97T^{2} \)
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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

−13.20562517273108978229664066351, −11.34622133274029039988953571775, −10.67019276252676296635009348463, −9.700904361232117750200167263973, −8.614666728887835615841036671452, −7.73703964893348709714163440024, −6.55314321322967584637318361014, −4.82491262168358477569632493973, −3.60800496623101173067285949740, −0.864208652357659300161169448728, 2.10376560199305166100303049316, 4.26184654181768534650831433465, 5.54109626339049705766674438727, 6.94874426170867693671852723273, 8.525512840279562471351607821652, 8.803122490443163463083083304378, 10.02120320190170304188941308070, 10.93703098429816637646947671019, 12.47673269856562940104771077002, 12.79183441380822445737485066841

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