Properties

Label 2-153-153.106-c1-0-11
Degree $2$
Conductor $153$
Sign $0.850 + 0.526i$
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  + (1.15 + 0.666i)2-s + (−0.864 − 1.50i)3-s + (−0.110 − 0.191i)4-s + (0.203 + 0.0545i)5-s + (0.00294 − 2.31i)6-s + (3.60 − 0.965i)7-s − 2.96i·8-s + (−1.50 + 2.59i)9-s + (0.198 + 0.198i)10-s + (−0.162 + 0.0436i)11-s + (−0.191 + 0.331i)12-s + (0.706 + 1.22i)13-s + (4.80 + 1.28i)14-s + (−0.0939 − 0.352i)15-s + (1.75 − 3.03i)16-s + (−4.12 + 0.0270i)17-s + ⋯
L(s)  = 1  + (0.816 + 0.471i)2-s + (−0.498 − 0.866i)3-s + (−0.0552 − 0.0957i)4-s + (0.0909 + 0.0243i)5-s + (0.00120 − 0.943i)6-s + (1.36 − 0.364i)7-s − 1.04i·8-s + (−0.502 + 0.864i)9-s + (0.0628 + 0.0628i)10-s + (−0.0490 + 0.0131i)11-s + (−0.0553 + 0.0956i)12-s + (0.196 + 0.339i)13-s + (1.28 + 0.344i)14-s + (−0.0242 − 0.0910i)15-s + (0.438 − 0.759i)16-s + (−0.999 + 0.00656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(153\)    =    \(3^{2} \cdot 17\)
Sign: $0.850 + 0.526i$
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: \((2,\ 153,\ (\ :1/2),\ 0.850 + 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41781 - 0.403715i\)
\(L(\frac12)\) \(\approx\) \(1.41781 - 0.403715i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.864 + 1.50i)T \)
17 \( 1 + (4.12 - 0.0270i)T \)
good2 \( 1 + (-1.15 - 0.666i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-0.203 - 0.0545i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-3.60 + 0.965i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (0.162 - 0.0436i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.706 - 1.22i)T + (-6.5 + 11.2i)T^{2} \)
19 \( 1 - 5.28iT - 19T^{2} \)
23 \( 1 + (1.82 - 6.79i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.863 - 3.22i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-6.79 - 1.82i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-6.83 + 6.83i)T - 37iT^{2} \)
41 \( 1 + (0.965 - 3.60i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (5.92 + 3.42i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.01 + 1.76i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6.04iT - 53T^{2} \)
59 \( 1 + (0.580 - 0.335i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.99 + 0.801i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.21 + 3.84i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.52 - 5.52i)T - 71iT^{2} \)
73 \( 1 + (4.07 - 4.07i)T - 73iT^{2} \)
79 \( 1 + (1.30 - 0.350i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-3.07 - 1.77i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 + (-2.32 - 8.66i)T + (-84.0 + 48.5i)T^{2} \)
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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

−13.16526204857393798762272023655, −11.99021634721283109933173331842, −11.20176502173296603197140684602, −10.02984058152598511300405578817, −8.352680654444127184506402946557, −7.37495638945078364420799309570, −6.25413927684057246592780377385, −5.29854271823713797091769832699, −4.20352424119458454476642102019, −1.62273885227308746995125545567, 2.61840889602926319507418438265, 4.36528514103055934762951228889, 4.86149656595930210963356693066, 6.13418754572589715894929799712, 8.103477197477648610185704421758, 8.934313956029627022480775732921, 10.40339703828450555140514791922, 11.44881654333369960631714916964, 11.71136666366620021015134194638, 13.05018803507763017698582569109

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