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Properties

Label 2-1904-476.191-c0-0-1
Degree $2$
Conductor $1904$
Sign $0.834 + 0.550i$
Analytic cond. $0.950219$
Root an. cond. $0.974792$
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  + (0.258 − 0.965i)3-s + (0.366 + 1.36i)5-s + (0.707 − 0.707i)7-s + (−0.965 − 0.258i)11-s − 13-s + 1.41·15-s + (0.866 − 0.5i)17-s + (0.707 − 1.22i)19-s + (−0.500 − 0.866i)21-s + (−0.866 + 0.5i)25-s + (0.707 − 0.707i)27-s + (1 − i)29-s + (−0.517 + 1.93i)31-s + (−0.499 + 0.866i)33-s + (1.22 + 0.707i)35-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)3-s + (0.366 + 1.36i)5-s + (0.707 − 0.707i)7-s + (−0.965 − 0.258i)11-s − 13-s + 1.41·15-s + (0.866 − 0.5i)17-s + (0.707 − 1.22i)19-s + (−0.500 − 0.866i)21-s + (−0.866 + 0.5i)25-s + (0.707 − 0.707i)27-s + (1 − i)29-s + (−0.517 + 1.93i)31-s + (−0.499 + 0.866i)33-s + (1.22 + 0.707i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(0.950219\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1904} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1904,\ (\ :0),\ 0.834 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.421861399\)
\(L(\frac12)\) \(\approx\) \(1.421861399\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
5 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-1 + i)T - iT^{2} \)
31 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + (-1 - i)T + iT^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
73 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
79 \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.430474160928843441855637565767, −8.195738259904977565455691118793, −7.40449066746964296718659526842, −7.30993249873815904036546887635, −6.44752193628854747038491178194, −5.33073329913973904758818493171, −4.51244683588579382396960443648, −2.85299408165881130464906974631, −2.67487236743316888053704767631, −1.24004226250255500724432688634, 1.45260850218096057634228639266, 2.58161704036015341718477187376, 3.86835255858916967228926960950, 4.71823252058634777636533993823, 5.31119023176779212851773508432, 5.79717260686729156260461618542, 7.47873422636363537817427305353, 8.034416802055997668016947181909, 8.853631033827840468610142055005, 9.462594181931665626488406835976

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