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Properties

Label 2-1900-380.303-c0-0-4
Degree $2$
Conductor $1900$
Sign $-0.973 - 0.229i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
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 + (−1.30 − 1.30i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 1.70i)6-s + (0.923 + 0.382i)8-s + 2.41i·9-s − 1.41i·11-s + 1.84·12-s + (1.30 − 1.30i)13-s − i·16-s + (2.23 − 0.923i)18-s + 19-s + (−1.30 + 0.541i)22-s + (−0.707 − 1.70i)24-s + (−1.70 − 0.707i)26-s + (1.84 − 1.84i)27-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−1.30 − 1.30i)3-s + (−0.707 + 0.707i)4-s + (−0.707 + 1.70i)6-s + (0.923 + 0.382i)8-s + 2.41i·9-s − 1.41i·11-s + 1.84·12-s + (1.30 − 1.30i)13-s − i·16-s + (2.23 − 0.923i)18-s + 19-s + (−1.30 + 0.541i)22-s + (−0.707 − 1.70i)24-s + (−1.70 − 0.707i)26-s + (1.84 − 1.84i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5428374929\)
\(L(\frac12)\) \(\approx\) \(0.5428374929\)
\(L(1)\) 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 \( 1 - T \)
good3 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
13 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
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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

−8.820495533903560994052642112998, −8.170655604688895261948583111744, −7.57846433849887001016489910264, −6.57523839562193699588624570310, −5.65749219634751646908656091491, −5.27866758275300040494709236093, −3.73220990931999586839118778391, −2.80330009239656723042943288687, −1.40899582786016670571648708225, −0.65401918380539009953019028975, 1.41162766731036922469977169289, 3.70952165612971071823472358277, 4.42103696348227727680075680467, 5.02270155842173468645848244939, 5.84426513795231623111247926015, 6.58491105492766475116640488527, 7.15986461244323053876872965652, 8.402915456919060621523404008213, 9.378385113773189503585689094019, 9.598504483326141121893887298845

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