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Properties

Label 2-1840-460.259-c0-0-0
Degree $2$
Conductor $1840$
Sign $-0.985 + 0.167i$
Analytic cond. $0.918279$
Root an. cond. $0.958269$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.74 − 0.512i)3-s + (−0.841 + 0.540i)5-s + (−0.281 + 1.95i)7-s + (1.94 + 1.24i)9-s + (1.74 − 0.512i)15-s + (1.49 − 3.27i)21-s + (−0.909 − 0.415i)23-s + (0.415 − 0.909i)25-s + (−1.56 − 1.80i)27-s + (−1.10 + 1.27i)29-s + (−0.822 − 1.80i)35-s + (0.239 − 0.153i)41-s + (1.45 + 0.425i)43-s − 2.30·45-s − 1.08·47-s + ⋯
L(s)  = 1  + (−1.74 − 0.512i)3-s + (−0.841 + 0.540i)5-s + (−0.281 + 1.95i)7-s + (1.94 + 1.24i)9-s + (1.74 − 0.512i)15-s + (1.49 − 3.27i)21-s + (−0.909 − 0.415i)23-s + (0.415 − 0.909i)25-s + (−1.56 − 1.80i)27-s + (−1.10 + 1.27i)29-s + (−0.822 − 1.80i)35-s + (0.239 − 0.153i)41-s + (1.45 + 0.425i)43-s − 2.30·45-s − 1.08·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.985 + 0.167i$
Analytic conductor: \(0.918279\)
Root analytic conductor: \(0.958269\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :0),\ -0.985 + 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1428554244\)
\(L(\frac12)\) \(\approx\) \(0.1428554244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.841 - 0.540i)T \)
23 \( 1 + (0.909 + 0.415i)T \)
good3 \( 1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2} \)
7 \( 1 + (0.281 - 1.95i)T + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (0.654 - 0.755i)T^{2} \)
13 \( 1 + (0.959 - 0.281i)T^{2} \)
17 \( 1 + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.142 - 0.989i)T^{2} \)
29 \( 1 + (1.10 - 1.27i)T + (-0.142 - 0.989i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 + (-0.415 - 0.909i)T^{2} \)
41 \( 1 + (-0.239 + 0.153i)T + (0.415 - 0.909i)T^{2} \)
43 \( 1 + (-1.45 - 0.425i)T + (0.841 + 0.540i)T^{2} \)
47 \( 1 + 1.08T + T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (0.959 - 0.281i)T^{2} \)
61 \( 1 + (1.25 - 0.368i)T + (0.841 - 0.540i)T^{2} \)
67 \( 1 + (-0.234 + 0.512i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (0.654 + 0.755i)T^{2} \)
73 \( 1 + (0.142 - 0.989i)T^{2} \)
79 \( 1 + (0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.474 + 0.304i)T + (0.415 + 0.909i)T^{2} \)
89 \( 1 + (0.797 + 0.234i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.415 + 0.909i)T^{2} \)
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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

−10.00722037095184164556856404721, −9.073025606471114084208164599275, −8.136252537140046798444541424263, −7.30343021011773000265809903498, −6.48467888225060727711119204864, −5.90353519404786408384424206149, −5.29186821651190867651121249533, −4.33511827833428649836055209594, −2.98439681571865976836951096283, −1.82418078621202756850036121260, 0.15091117750522916914612637152, 1.20661144017099657544520538219, 3.73344735520832400290823672621, 4.10796528144422877680007721159, 4.80477277384398981298309725105, 5.77348386717945836827218360506, 6.58951293880355846849527491689, 7.39764038203618053389666209081, 7.902295320114693900983854269017, 9.400476107257666736430696470698

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