Properties

Label 2-154-7.4-c1-0-2
Degree $2$
Conductor $154$
Sign $-0.266 - 0.963i$
Analytic cond. $1.22969$
Root an. cond. $1.10891$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 0.999·6-s + (−0.5 + 2.59i)7-s − 0.999·8-s + (1 + 1.73i)9-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + (−2.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (3 − 5.19i)17-s + (−1 + 1.73i)18-s + (−1 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.408·6-s + (−0.188 + 0.981i)7-s − 0.353·8-s + (0.333 + 0.577i)9-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s − 0.277·13-s + (−0.668 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (0.727 − 1.26i)17-s + (−0.235 + 0.408i)18-s + (−0.229 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713254 + 0.937560i\)
\(L(\frac12)\) \(\approx\) \(0.713254 + 0.937560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 2.59i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.5 + 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1 - 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-9 - 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−13.38765584262575967164939491687, −12.25300655191903931582520382412, −11.45031183053326392703775854309, −10.09941733149404299284967439672, −9.147667252246725439215261619985, −7.987608136448294360361605160408, −6.74919898347745185779226083490, −5.47380884793701625154904166440, −4.67749603755539734972635170355, −2.90298281088910759211246479290, 1.27264219928552330736687837556, 3.40638345820900232063903582841, 4.64334491609587407803611653671, 6.26055009210381841320856558310, 7.13501809022469655878000348846, 8.580872803062291447603636051153, 10.04865144348258262782234734684, 10.57189139206522898329107625346, 11.96112863613740154824481702254, 12.57211878491947900670966175788

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