Properties

Label 2-154-7.4-c1-0-4
Degree $2$
Conductor $154$
Sign $0.991 - 0.126i$
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 + (1.5 − 2.59i)3-s + (−0.499 + 0.866i)4-s + (2 + 3.46i)5-s + 3·6-s + (−2.5 − 0.866i)7-s − 0.999·8-s + (−3 − 5.19i)9-s + (−1.99 + 3.46i)10-s + (0.5 − 0.866i)11-s + (1.50 + 2.59i)12-s − 13-s + (−0.500 − 2.59i)14-s + 12·15-s + (−0.5 − 0.866i)16-s + (−1 + 1.73i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 − 1.49i)3-s + (−0.249 + 0.433i)4-s + (0.894 + 1.54i)5-s + 1.22·6-s + (−0.944 − 0.327i)7-s − 0.353·8-s + (−1 − 1.73i)9-s + (−0.632 + 1.09i)10-s + (0.150 − 0.261i)11-s + (0.433 + 0.749i)12-s − 0.277·13-s + (−0.133 − 0.694i)14-s + 3.09·15-s + (−0.125 − 0.216i)16-s + (−0.242 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(154\)    =    \(2 \cdot 7 \cdot 11\)
Sign: $0.991 - 0.126i$
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: \((2,\ 154,\ (\ :1/2),\ 0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65038 + 0.104733i\)
\(L(\frac12)\) \(\approx\) \(1.65038 + 0.104733i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 + 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
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 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.5 + 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.5 - 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5T + 97T^{2} \)
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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

−13.38846371560432672660405971099, −12.51574387827071077619440490772, −11.04623671965480280605363191019, −9.735594596058672393343829737667, −8.617941701912037415993615121931, −7.17433214611602209685784023249, −6.82500268444453572766359887593, −6.01246055865774076949142190835, −3.35822139059151646624000002223, −2.39971980369470025644422909823, 2.35960317471923201559308793654, 3.90545201001924950217398459583, 4.85949936813717926247574447694, 5.89569493697794276035855828275, 8.406253260525527358347099467565, 9.286266462495648396467298288701, 9.664363201484915324423128282110, 10.55333383450835637937244143279, 12.20541195663546005208323134745, 12.96528287315446072102700425506

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