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Properties

Label 2-3800-152.37-c0-0-15
Degree $2$
Conductor $3800$
Sign $0.382 + 0.923i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
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  + (0.382 − 0.923i)2-s + 1.84·3-s + (−0.707 − 0.707i)4-s + (0.707 − 1.70i)6-s + (−0.923 + 0.382i)8-s + 2.41·9-s + 1.41i·11-s + (−1.30 − 1.30i)12-s + 0.765·13-s + i·16-s + (0.923 − 2.23i)18-s − i·19-s + (1.30 + 0.541i)22-s + (−1.70 + 0.707i)24-s + (0.292 − 0.707i)26-s + 2.61·27-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + 1.84·3-s + (−0.707 − 0.707i)4-s + (0.707 − 1.70i)6-s + (−0.923 + 0.382i)8-s + 2.41·9-s + 1.41i·11-s + (−1.30 − 1.30i)12-s + 0.765·13-s + i·16-s + (0.923 − 2.23i)18-s − i·19-s + (1.30 + 0.541i)22-s + (−1.70 + 0.707i)24-s + (0.292 − 0.707i)26-s + 2.61·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.830887617\)
\(L(\frac12)\) \(\approx\) \(2.830887617\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 \)
19 \( 1 + iT \)
good3 \( 1 - 1.84T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - 0.765T + T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 1.84T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.84T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 1.41iT - T^{2} \)
67 \( 1 - 0.765T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - 0.765iT - T^{2} \)
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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

−8.638301129172775102737270931588, −8.113637942878325027280875805355, −7.17018294709930161992722496362, −6.50499874624353533460016946167, −5.06255089721723305546171957829, −4.48059131645615754939940392084, −3.64473176446447483324545418151, −3.04453720770457470568375216167, −2.11982514932773217352444826942, −1.54463113061535235245841511351, 1.50711970670807164268213917037, 2.85398583604329477404047887834, 3.50738307903992242220646157842, 3.91965350346931360812277424888, 5.02551103126484873641973915024, 5.99660213745261177032376074684, 6.68394701764801132248834245649, 7.58094127778790936452492461395, 8.159407116191031498226739870339, 8.595911731452791349553838536898

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