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Properties

Label 2-783-9.7-c1-0-1
Degree $2$
Conductor $783$
Sign $-0.999 + 0.0178i$
Analytic cond. $6.25228$
Root an. cond. $2.50045$
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  + (−0.489 + 0.847i)2-s + (0.520 + 0.902i)4-s + (1.17 + 2.03i)5-s + (0.172 − 0.299i)7-s − 2.97·8-s − 2.29·10-s + (−2.90 + 5.02i)11-s + (−1.50 − 2.61i)13-s + (0.169 + 0.292i)14-s + (0.415 − 0.719i)16-s − 1.63·17-s − 1.61·19-s + (−1.22 + 2.11i)20-s + (−2.84 − 4.92i)22-s + (−0.565 − 0.980i)23-s + ⋯
L(s)  = 1  + (−0.346 + 0.599i)2-s + (0.260 + 0.451i)4-s + (0.524 + 0.908i)5-s + (0.0652 − 0.113i)7-s − 1.05·8-s − 0.725·10-s + (−0.875 + 1.51i)11-s + (−0.418 − 0.724i)13-s + (0.0451 + 0.0782i)14-s + (0.103 − 0.179i)16-s − 0.397·17-s − 0.369·19-s + (−0.273 + 0.473i)20-s + (−0.606 − 1.04i)22-s + (−0.117 − 0.204i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0178i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $-0.999 + 0.0178i$
Analytic conductor: \(6.25228\)
Root analytic conductor: \(2.50045\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (262, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 783,\ (\ :1/2),\ -0.999 + 0.0178i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00904231 - 1.01346i\)
\(L(\frac12)\) \(\approx\) \(0.00904231 - 1.01346i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.489 - 0.847i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-1.17 - 2.03i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.172 + 0.299i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.90 - 5.02i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.50 + 2.61i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.63T + 17T^{2} \)
19 \( 1 + 1.61T + 19T^{2} \)
23 \( 1 + (0.565 + 0.980i)T + (-11.5 + 19.9i)T^{2} \)
31 \( 1 + (-3.73 - 6.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.72T + 37T^{2} \)
41 \( 1 + (-4.41 - 7.64i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.422 + 0.732i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.93 + 6.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 4.90T + 53T^{2} \)
59 \( 1 + (3.35 + 5.80i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.07 + 3.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.24 - 5.62i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.1T + 71T^{2} \)
73 \( 1 + 12.1T + 73T^{2} \)
79 \( 1 + (5.91 - 10.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.06 - 10.5i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.10T + 89T^{2} \)
97 \( 1 + (-1.79 + 3.11i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36986390072362104964674594396, −10.07604837615583641487051528056, −8.894026393735676906085185275173, −7.959841885080931678224214289409, −7.20060409632975955106342747176, −6.68939170136545826411461149492, −5.63900346085194790162150899255, −4.49178374121919874000181913615, −2.97711116155671761816442544396, −2.27292881257918723212184339063, 0.52737342584639698776748414563, 1.88853609340467996394216435686, 2.89537783994326431676600255839, 4.42369436453760917542010990688, 5.63029188210405770514006702389, 5.98687251195121913114601448810, 7.34504626824495416351696504230, 8.650147993900839206564730826744, 8.967410887468107080837874442137, 9.930678977717751660504457463523

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