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Properties

Label 2-845-65.18-c1-0-62
Degree $2$
Conductor $845$
Sign $-0.638 - 0.769i$
Analytic cond. $6.74735$
Root an. cond. $2.59756$
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  − 1.58i·2-s + (−0.139 − 0.139i)3-s − 0.511·4-s + (−2.23 − 0.0672i)5-s + (−0.221 + 0.221i)6-s − 0.548·7-s − 2.35i·8-s − 2.96i·9-s + (−0.106 + 3.54i)10-s + (−0.108 − 0.108i)11-s + (0.0713 + 0.0713i)12-s + 0.868i·14-s + (0.302 + 0.321i)15-s − 4.76·16-s + (2.22 + 2.22i)17-s − 4.69·18-s + ⋯
L(s)  = 1  − 1.12i·2-s + (−0.0805 − 0.0805i)3-s − 0.255·4-s + (−0.999 − 0.0300i)5-s + (−0.0902 + 0.0902i)6-s − 0.207·7-s − 0.834i·8-s − 0.987i·9-s + (−0.0337 + 1.12i)10-s + (−0.0326 − 0.0326i)11-s + (0.0205 + 0.0205i)12-s + 0.232i·14-s + (0.0780 + 0.0829i)15-s − 1.19·16-s + (0.539 + 0.539i)17-s − 1.10·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(845\)    =    \(5 \cdot 13^{2}\)
Sign: $-0.638 - 0.769i$
Analytic conductor: \(6.74735\)
Root analytic conductor: \(2.59756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{845} (408, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 845,\ (\ :1/2),\ -0.638 - 0.769i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.258375 + 0.550492i\)
\(L(\frac12)\) \(\approx\) \(0.258375 + 0.550492i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.0672i)T \)
13 \( 1 \)
good2 \( 1 + 1.58iT - 2T^{2} \)
3 \( 1 + (0.139 + 0.139i)T + 3iT^{2} \)
7 \( 1 + 0.548T + 7T^{2} \)
11 \( 1 + (0.108 + 0.108i)T + 11iT^{2} \)
17 \( 1 + (-2.22 - 2.22i)T + 17iT^{2} \)
19 \( 1 + (3.22 + 3.22i)T + 19iT^{2} \)
23 \( 1 + (2.50 - 2.50i)T - 23iT^{2} \)
29 \( 1 + 2.34iT - 29T^{2} \)
31 \( 1 + (6.60 - 6.60i)T - 31iT^{2} \)
37 \( 1 + 6.80T + 37T^{2} \)
41 \( 1 + (-2.53 + 2.53i)T - 41iT^{2} \)
43 \( 1 + (5.02 - 5.02i)T - 43iT^{2} \)
47 \( 1 + 9.13T + 47T^{2} \)
53 \( 1 + (3.70 + 3.70i)T + 53iT^{2} \)
59 \( 1 + (-2.69 + 2.69i)T - 59iT^{2} \)
61 \( 1 - 7.84T + 61T^{2} \)
67 \( 1 + 4.89iT - 67T^{2} \)
71 \( 1 + (-11.0 + 11.0i)T - 71iT^{2} \)
73 \( 1 - 3.91iT - 73T^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 - 13.4T + 83T^{2} \)
89 \( 1 + (-6.43 + 6.43i)T - 89iT^{2} \)
97 \( 1 - 7.57iT - 97T^{2} \)
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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

−9.813531517488949833773241159383, −9.022384412816950880108084249566, −8.073926548400572392854488028879, −6.98949670456398765665117019247, −6.35498362077324199165232437139, −4.86726397109504855834782859789, −3.64698611538687796641773578112, −3.32107051303554426607471307149, −1.73908522101519657724997704677, −0.28607700011836006863388936583, 2.18523734740445213715844425393, 3.62104387301692896998334853136, 4.75848074565500040057226147084, 5.52699459590328034461269974809, 6.57956571689878017282733165414, 7.36266807977105522439502174332, 8.047759483821567392365443218194, 8.562644107768739774515673174767, 9.852027598829205780719231582183, 10.82799506168554732527381515586

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