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Properties

Label 2-154-11.4-c1-0-1
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  + (0.809 + 0.587i)2-s + (−0.5 + 1.53i)3-s + (0.309 + 0.951i)4-s + (−1.61 + 1.17i)5-s + (−1.30 + 0.951i)6-s + (−0.309 − 0.951i)7-s + (−0.309 + 0.951i)8-s + (0.309 + 0.224i)9-s − 2·10-s + (2.54 + 2.12i)11-s − 1.61·12-s + (−2.61 − 1.90i)13-s + (0.309 − 0.951i)14-s + (−1 − 3.07i)15-s + (−0.809 + 0.587i)16-s + (4.73 − 3.44i)17-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.288 + 0.888i)3-s + (0.154 + 0.475i)4-s + (−0.723 + 0.525i)5-s + (−0.534 + 0.388i)6-s + (−0.116 − 0.359i)7-s + (−0.109 + 0.336i)8-s + (0.103 + 0.0748i)9-s − 0.632·10-s + (0.767 + 0.641i)11-s − 0.467·12-s + (−0.726 − 0.527i)13-s + (0.0825 − 0.254i)14-s + (−0.258 − 0.794i)15-s + (−0.202 + 0.146i)16-s + (1.14 − 0.834i)17-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: $\chi_{154} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.767394 + 1.00882i\)
\(L(\frac12)\) \(\approx\) \(0.767394 + 1.00882i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 + (-2.54 - 2.12i)T \)
good3 \( 1 + (0.5 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.61 - 1.17i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.61 + 1.90i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.73 + 3.44i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.0450 - 0.138i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.47T + 23T^{2} \)
29 \( 1 + (2.47 + 7.60i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.61 - 2.62i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.38 + 7.33i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.263 + 0.812i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.85T + 43T^{2} \)
47 \( 1 + (-0.527 + 1.62i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4 + 2.90i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.28 - 3.94i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.23 - 3.80i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.09T + 67T^{2} \)
71 \( 1 + (8.47 - 6.15i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.736 + 2.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-1 - 0.726i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.5 + 7.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 + (-2.54 - 1.84i)T + (29.9 + 92.2i)T^{2} \)
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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

−13.31952980119216205737397991064, −12.17599666463807501266493703275, −11.34339642369084811111952450588, −10.28681246964921251286306149157, −9.363305740797403209411301471914, −7.62241219327127254151612703984, −7.02903736386470540356683366716, −5.35675389576477335025645656805, −4.35940361262738733449306170345, −3.25055728439714685284170448714, 1.30087587556126454276532233934, 3.38923474694664569271810488696, 4.80763232797740526872568148057, 6.17682651028164146038573420447, 7.18146155779918971471294478697, 8.461459176239796518942519632572, 9.644149428177363340620510662035, 11.11036314167889103298545493998, 12.07204625866992238070464996868, 12.40355775918381762661517571215

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