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Properties

Label 2-154-7.2-c1-0-0
Degree $2$
Conductor $154$
Sign $-0.198 - 0.980i$
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 + (1.20 + 2.09i)3-s + (−0.499 − 0.866i)4-s + (0.292 − 0.507i)5-s − 2.41·6-s + (1.62 + 2.09i)7-s + 0.999·8-s + (−1.41 + 2.44i)9-s + (0.292 + 0.507i)10-s + (−0.5 − 0.866i)11-s + (1.20 − 2.09i)12-s − 3.82·13-s + (−2.62 + 0.358i)14-s + 1.41·15-s + (−0.5 + 0.866i)16-s + (−1.82 − 3.16i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.696 + 1.20i)3-s + (−0.249 − 0.433i)4-s + (0.130 − 0.226i)5-s − 0.985·6-s + (0.612 + 0.790i)7-s + 0.353·8-s + (−0.471 + 0.816i)9-s + (0.0926 + 0.160i)10-s + (−0.150 − 0.261i)11-s + (0.348 − 0.603i)12-s − 1.06·13-s + (−0.700 + 0.0958i)14-s + 0.365·15-s + (−0.125 + 0.216i)16-s + (−0.443 − 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $-0.198 - 0.980i$
Analytic conductor: \(1.22969\)
Root analytic conductor: \(1.10891\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{154} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.740450 + 0.905031i\)
\(L(\frac12)\) \(\approx\) \(0.740450 + 0.905031i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 2.09i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.20 - 2.09i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.292 + 0.507i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.12 + 5.40i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.70 + 8.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5.41T + 41T^{2} \)
43 \( 1 + 5.65T + 43T^{2} \)
47 \( 1 + (-5.24 + 9.08i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.94 + 6.84i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.79 - 4.83i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.91 - 10.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.37 + 2.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.0T + 71T^{2} \)
73 \( 1 + (-4.70 - 8.15i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.62 + 11.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (6.24 - 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−13.59894850164039312020394301925, −12.20888650071588788742632131686, −10.92559497435590206325080887873, −9.903151453491539941925150321330, −9.004304191124823614415120313252, −8.463570247834292851715321678827, −7.04878483542018851296720419184, −5.31643281918903787146830163960, −4.59308254135372587184593459195, −2.71524795317829369046764101724, 1.52392622082547512492442252022, 2.83541669813597376153202499221, 4.60758829445336349660510693597, 6.68205235612353318408213383351, 7.63430176786252485910067063432, 8.299257505895437554564725469101, 9.666856748725771587468772728471, 10.67432880360600039918418344422, 11.79867477640957568550949137285, 12.74191238863151099607806074505

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