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Properties

Label 2-819-117.103-c1-0-21
Degree $2$
Conductor $819$
Sign $0.840 - 0.541i$
Analytic cond. $6.53974$
Root an. cond. $2.55729$
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  + (1.12 − 0.650i)2-s + (−1.21 − 1.23i)3-s + (−0.153 + 0.265i)4-s + (0.219 + 0.126i)5-s + (−2.17 − 0.597i)6-s + (0.866 − 0.5i)7-s + 3.00i·8-s + (−0.0382 + 2.99i)9-s + 0.330·10-s + (−1.66 + 0.961i)11-s + (0.513 − 0.134i)12-s + (−2.14 + 2.89i)13-s + (0.650 − 1.12i)14-s + (−0.111 − 0.425i)15-s + (1.64 + 2.85i)16-s + 5.23·17-s + ⋯
L(s)  = 1  + (0.796 − 0.460i)2-s + (−0.702 − 0.711i)3-s + (−0.0766 + 0.132i)4-s + (0.0983 + 0.0567i)5-s + (−0.887 − 0.243i)6-s + (0.327 − 0.188i)7-s + 1.06i·8-s + (−0.0127 + 0.999i)9-s + 0.104·10-s + (−0.502 + 0.289i)11-s + (0.148 − 0.0387i)12-s + (−0.594 + 0.803i)13-s + (0.173 − 0.301i)14-s + (−0.0286 − 0.109i)15-s + (0.411 + 0.713i)16-s + 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 - 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.840 - 0.541i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.840 - 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46508 + 0.430768i\)
\(L(\frac12)\) \(\approx\) \(1.46508 + 0.430768i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.21 + 1.23i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (2.14 - 2.89i)T \)
good2 \( 1 + (-1.12 + 0.650i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.219 - 0.126i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.66 - 0.961i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 - 5.23T + 17T^{2} \)
19 \( 1 - 2.96iT - 19T^{2} \)
23 \( 1 + (2.95 - 5.11i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.23 - 2.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.78 - 5.07i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 1.00iT - 37T^{2} \)
41 \( 1 + (1.53 + 0.886i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.26 - 2.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.74 + 2.16i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.06T + 53T^{2} \)
59 \( 1 + (-9.78 - 5.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.98 + 12.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.8 - 6.83i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.99iT - 71T^{2} \)
73 \( 1 + 5.60iT - 73T^{2} \)
79 \( 1 + (4.42 + 7.67i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.83 - 5.09i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 17.4iT - 89T^{2} \)
97 \( 1 + (-1.53 + 0.887i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.49347825593379604184548393919, −9.778856515832619931342971745199, −8.236880763699278526928189250128, −7.80176508342018061401189657558, −6.76743890101508164971940749674, −5.65316482026024882476291330216, −4.99320445069177231514191667078, −4.03791775210026454991214041511, −2.70587059853601192037377033465, −1.59687542389339083557749291672, 0.66562501359732825407827090514, 2.91843258900464963104170233432, 4.12138160150376194205302210690, 4.95807954921753821111843532311, 5.60229832096919751385123060682, 6.23224633734479642649910181028, 7.39664276092862813066482352808, 8.412766950399361106351633456494, 9.704165890128660553472021890468, 10.03785962003157387405657652590

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