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Properties

Label 2-154-7.4-c1-0-5
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 $1$

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 + (−1 − 1.73i)5-s + 3·6-s + (−2.5 − 0.866i)7-s + 0.999·8-s + (−3 − 5.19i)9-s + (−0.999 + 1.73i)10-s + (0.5 − 0.866i)11-s + (−1.50 − 2.59i)12-s − 7·13-s + (0.500 + 2.59i)14-s + 6·15-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 + (−0.447 − 0.774i)5-s + 1.22·6-s + (−0.944 − 0.327i)7-s + 0.353·8-s + (−1 − 1.73i)9-s + (−0.316 + 0.547i)10-s + (0.150 − 0.261i)11-s + (−0.433 − 0.749i)12-s − 1.94·13-s + (0.133 + 0.694i)14-s + 1.54·15-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: \(1\)
Selberg data: \((2,\ 154,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 7T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (2 - 3.46i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−12.19101119110010568853914852527, −11.39582531344358494744111048022, −10.33890855853346609793653845088, −9.640054305598878299594123767399, −8.919028817672611635365238972700, −7.21464329488200494738000024095, −5.46718379167535920617644103564, −4.48712644075065081362246790571, −3.39877748303546376643881966778, 0, 2.53903065802978355823610921284, 5.09637708040510942379325153813, 6.42045675232671820976792212730, 7.03280613820422951654737666199, 7.67646855496041689654652526508, 9.272444616911073845565655902315, 10.50739351479852724599811809410, 11.64980475898635615359020610186, 12.46577484690440423303006952011

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