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Properties

Label 2-154-7.2-c1-0-1
Degree $2$
Conductor $154$
Sign $0.832 - 0.553i$
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 + (1.32 + 2.29i)3-s + (−0.499 − 0.866i)4-s + (−1.82 + 3.15i)5-s + 2.64·6-s + (1.32 − 2.29i)7-s − 0.999·8-s + (−2 + 3.46i)9-s + (1.82 + 3.15i)10-s + (−0.5 − 0.866i)11-s + (1.32 − 2.29i)12-s + 5·13-s + (−1.32 − 2.29i)14-s − 9.64·15-s + (−0.5 + 0.866i)16-s + (−3 − 5.19i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.763 + 1.32i)3-s + (−0.249 − 0.433i)4-s + (−0.815 + 1.41i)5-s + 1.08·6-s + (0.499 − 0.866i)7-s − 0.353·8-s + (−0.666 + 1.15i)9-s + (0.576 + 0.998i)10-s + (−0.150 − 0.261i)11-s + (0.381 − 0.661i)12-s + 1.38·13-s + (−0.353 − 0.612i)14-s − 2.49·15-s + (−0.125 + 0.216i)16-s + (−0.727 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.832 - 0.553i$
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: \((2,\ 154,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42258 + 0.429926i\)
\(L(\frac12)\) \(\approx\) \(1.42258 + 0.429926i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 - 5T + 13T^{2} \)
17 \( 1 + (3 + 5.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.177 + 0.306i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.82 + 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.29T + 29T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.822 + 1.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (6.64 - 11.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.82 - 3.15i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.322 - 0.559i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.85 - 3.21i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.96 + 3.40i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.64T + 71T^{2} \)
73 \( 1 + (2.82 + 4.88i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.32 - 2.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 13.2T + 83T^{2} \)
89 \( 1 + (-7.29 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.70T + 97T^{2} \)
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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

−13.44886311356135197080926410369, −11.51981681984956004577269622171, −10.90890385957543535262720454839, −10.42444873650104139226641275556, −9.169116142552141687309255412640, −8.021986884520513601419562775696, −6.69797574880814778690093801995, −4.70840771771047264601916013173, −3.73725266090806095837061049493, −2.96298327526848233382819988489, 1.68186501093571855182183849520, 3.79638292825034123679365534386, 5.26612020791751810280609560819, 6.53744348665975916969101268034, 7.968035872748558359004354378445, 8.346903938502269848788683500109, 9.034082311170186691341034692542, 11.42574412228626688371640984991, 12.24950216509368309784013023516, 13.12628759791328036638152222017

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