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Properties

Label 2-468-117.25-c1-0-6
Degree $2$
Conductor $468$
Sign $0.993 + 0.109i$
Analytic cond. $3.73699$
Root an. cond. $1.93313$
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  + (1.5 − 0.866i)3-s + (−1.73 + i)5-s + (1.73 + i)7-s + (1.5 − 2.59i)9-s + (5.19 + 3i)11-s + (−3.23 − 1.59i)13-s + (−1.73 + 3i)15-s + 5·17-s − 6i·19-s + 3.46·21-s + (3.5 + 6.06i)23-s + (−0.500 + 0.866i)25-s − 5.19i·27-s + (1 − 1.73i)29-s + (−1.73 + i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.774 + 0.447i)5-s + (0.654 + 0.377i)7-s + (0.5 − 0.866i)9-s + (1.56 + 0.904i)11-s + (−0.896 − 0.443i)13-s + (−0.447 + 0.774i)15-s + 1.21·17-s − 1.37i·19-s + 0.755·21-s + (0.729 + 1.26i)23-s + (−0.100 + 0.173i)25-s − 0.999i·27-s + (0.185 − 0.321i)29-s + (−0.311 + 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.993 + 0.109i$
Analytic conductor: \(3.73699\)
Root analytic conductor: \(1.93313\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :1/2),\ 0.993 + 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87265 - 0.103221i\)
\(L(\frac12)\) \(\approx\) \(1.87265 - 0.103221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
13 \( 1 + (3.23 + 1.59i)T \)
good5 \( 1 + (1.73 - i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-5.19 - 3i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (-3.5 - 6.06i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1 + 1.73i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.73 - i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + (3.46 - 2i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.3 + 6i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 3T + 53T^{2} \)
59 \( 1 + (8.66 - 5i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.8 + 8i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + (1.5 - 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 6iT - 89T^{2} \)
97 \( 1 + (10.3 + 6i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.33443772168481727898807806010, −9.820132662788391203602865725292, −9.236314687196555139135523584515, −8.172263018858299848971878812173, −7.32417629933033040134513029310, −6.86037241358473885413147670240, −5.20099525034466483901487050208, −3.96893338787545051340815679694, −2.95447991201873568136585590776, −1.52721857936614427513404751413, 1.44481820233298870482583408566, 3.29464639161079180600690224435, 4.12567033656532418794037552192, 4.97528399026807937808425878234, 6.53145236908709413264316123138, 7.78584719214947219705173386876, 8.265765377738218319053239283737, 9.170604968442252445991466055583, 10.02869248991817616114919114177, 11.04291733271368803145917334719

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