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Properties

Label 2-819-91.23-c1-0-34
Degree $2$
Conductor $819$
Sign $0.0782 + 0.996i$
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  + 1.78i·2-s − 1.19·4-s + (−0.972 − 0.561i)5-s + (−2.33 + 1.24i)7-s + 1.44i·8-s + (1.00 − 1.73i)10-s + (−4.27 − 2.46i)11-s + (3.59 − 0.216i)13-s + (−2.22 − 4.16i)14-s − 4.96·16-s − 3.82·17-s + (2.23 − 1.29i)19-s + (1.15 + 0.668i)20-s + (4.40 − 7.62i)22-s − 1.14·23-s + ⋯
L(s)  = 1  + 1.26i·2-s − 0.595·4-s + (−0.434 − 0.251i)5-s + (−0.881 + 0.471i)7-s + 0.511i·8-s + (0.317 − 0.549i)10-s + (−1.28 − 0.743i)11-s + (0.998 − 0.0599i)13-s + (−0.595 − 1.11i)14-s − 1.24·16-s − 0.928·17-s + (0.513 − 0.296i)19-s + (0.258 + 0.149i)20-s + (0.939 − 1.62i)22-s − 0.238·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0782 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0782 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(819\)    =    \(3^{2} \cdot 7 \cdot 13\)
Sign: $0.0782 + 0.996i$
Analytic conductor: \(6.53974\)
Root analytic conductor: \(2.55729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{819} (478, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 819,\ (\ :1/2),\ 0.0782 + 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00858379 - 0.00793657i\)
\(L(\frac12)\) \(\approx\) \(0.00858379 - 0.00793657i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.33 - 1.24i)T \)
13 \( 1 + (-3.59 + 0.216i)T \)
good2 \( 1 - 1.78iT - 2T^{2} \)
5 \( 1 + (0.972 + 0.561i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (4.27 + 2.46i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 3.82T + 17T^{2} \)
19 \( 1 + (-2.23 + 1.29i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.14T + 23T^{2} \)
29 \( 1 + (0.502 + 0.869i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.73 - 2.15i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 + (5.39 - 3.11i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.22 + 2.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.50 + 3.75i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.30 + 7.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 + (4.09 + 7.08i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.58 - 3.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.35 - 5.39i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (9.63 - 5.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (8.30 - 14.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.75iT - 83T^{2} \)
89 \( 1 - 12.2iT - 89T^{2} \)
97 \( 1 + (-7.56 - 4.37i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.846702175764982990204449209655, −8.665591997877661445641267282052, −8.405375547903416068630399444353, −7.40203893912625726253279238592, −6.51190217509936450446456385801, −5.79066767868735031885151070483, −5.04013135457103540194914025417, −3.68124172544389473931881732750, −2.47801267392321120743694729343, −0.00536531517675284613454928194, 1.74595410411612481345842140774, 2.99841917132116927932278648797, 3.68050561924352154211326466339, 4.70568917224153233270870644161, 6.13169842954794015779395556331, 7.06664204672980257073018861463, 7.85849831782456918615448240168, 9.118962455754473422381122693997, 9.867479997748593965171145699565, 10.56643366806325480103666785670

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