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Properties

Label 2-3800-152.37-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(0.4133086609\)
\(L(\frac12)\) \(\approx\) \(0.4133086609\)
\(L(1)\) 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

−8.504655729019005346217703513311, −7.72809254991887763441794896864, −6.95838161732376364848602245355, −5.94515421005170350072166393019, −5.55287522359733585113632507757, −4.59566372591461259796922387648, −3.94749060311967316201114333561, −2.83156144406256263021252708944, −1.50148377331116799535529941165, −0.51044433845691587461890953830, 0.869058986565000773675567928839, 2.13493783075818219627071734273, 4.12349842887505980379877002231, 4.81784781246868365437143720671, 5.11918043372341302107863729853, 6.09997269660975271167861319084, 6.63394738995838425156368406983, 7.30222363296341418958668586178, 7.72817356946768780319933578032, 9.032499894065345255728359104420

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