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Properties

Label 2-819-91.51-c1-0-41
Degree $2$
Conductor $819$
Sign $-0.998 + 0.0556i$
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  + (0.287 + 0.166i)2-s + (−0.944 − 1.63i)4-s + (1.25 + 0.722i)5-s + (−2.26 + 1.36i)7-s − 1.29i·8-s + (0.240 + 0.416i)10-s + (−5.15 + 2.97i)11-s + (1.88 − 3.07i)13-s + (−0.879 + 0.0178i)14-s + (−1.67 + 2.90i)16-s + (−2.16 − 3.74i)17-s + (−1.69 − 0.978i)19-s − 2.73i·20-s − 1.97·22-s + (0.270 − 0.467i)23-s + ⋯
L(s)  = 1  + (0.203 + 0.117i)2-s + (−0.472 − 0.818i)4-s + (0.559 + 0.323i)5-s + (−0.855 + 0.517i)7-s − 0.457i·8-s + (0.0759 + 0.131i)10-s + (−1.55 + 0.897i)11-s + (0.524 − 0.851i)13-s + (−0.234 + 0.00477i)14-s + (−0.418 + 0.725i)16-s + (−0.524 − 0.909i)17-s + (−0.388 − 0.224i)19-s − 0.610i·20-s − 0.421·22-s + (0.0563 − 0.0975i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $-0.998 + 0.0556i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ -0.998 + 0.0556i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00349617 - 0.125499i\)
\(L(\frac12)\) \(\approx\) \(0.00349617 - 0.125499i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (2.26 - 1.36i)T \)
13 \( 1 + (-1.88 + 3.07i)T \)
good2 \( 1 + (-0.287 - 0.166i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + (-1.25 - 0.722i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (5.15 - 2.97i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.16 + 3.74i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.69 + 0.978i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.270 + 0.467i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 7.15T + 29T^{2} \)
31 \( 1 + (5.28 - 3.05i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.95 - 4.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.55iT - 41T^{2} \)
43 \( 1 + 4.24T + 43T^{2} \)
47 \( 1 + (5.42 + 3.13i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.38 + 2.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.737 - 0.425i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.38 - 5.87i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.854 - 0.493i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.76iT - 71T^{2} \)
73 \( 1 + (-7.91 + 4.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.0655 + 0.113i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.66iT - 83T^{2} \)
89 \( 1 + (8.41 + 4.85i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 6.58iT - 97T^{2} \)
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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

−9.861091983026582767318485040880, −9.250954556808602841092338444848, −8.158599874383664872290130814592, −7.03218113592651134267085126659, −6.14387416155852184483772442198, −5.43815606074526119475735414507, −4.65294289480294931391308232496, −3.12747776159854682204193590751, −2.10231726806498360396574219695, −0.05330349506837263551136377576, 2.13757921612393231065117900333, 3.41656150145756319584594548050, 4.11944851696746892600672192258, 5.39139277454279166105469291259, 6.14363857557120896806085775816, 7.35105252002018092154978742458, 8.146835406302598965371120403311, 9.010553682039015722818603230604, 9.645854910496740090796673178376, 10.79815178771924009257855187498

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