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Properties

Label 2-1904-1904.237-c0-0-2
Degree $2$
Conductor $1904$
Sign $-0.996 + 0.0784i$
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  + (−0.951 + 0.309i)2-s + (1.26 + 1.26i)3-s + (0.809 − 0.587i)4-s + (−1.39 + 1.39i)5-s + (−1.58 − 0.809i)6-s + i·7-s + (−0.587 + 0.809i)8-s + 2.17i·9-s + (0.896 − 1.76i)10-s + (1.76 + 0.278i)12-s + (−0.309 − 0.951i)14-s − 3.52·15-s + (0.309 − 0.951i)16-s + 17-s + (−0.672 − 2.06i)18-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (1.26 + 1.26i)3-s + (0.809 − 0.587i)4-s + (−1.39 + 1.39i)5-s + (−1.58 − 0.809i)6-s + i·7-s + (−0.587 + 0.809i)8-s + 2.17i·9-s + (0.896 − 1.76i)10-s + (1.76 + 0.278i)12-s + (−0.309 − 0.951i)14-s − 3.52·15-s + (0.309 − 0.951i)16-s + 17-s + (−0.672 − 2.06i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $-0.996 + 0.0784i$
Analytic conductor: \(0.950219\)
Root analytic conductor: \(0.974792\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1904} (237, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1904,\ (\ :0),\ -0.996 + 0.0784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8892722906\)
\(L(\frac12)\) \(\approx\) \(0.8892722906\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
7 \( 1 - iT \)
17 \( 1 - T \)
good3 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
5 \( 1 + (1.39 - 1.39i)T - iT^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - 1.61T + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 0.618iT - T^{2} \)
43 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.221 + 0.221i)T - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1.26 + 1.26i)T + iT^{2} \)
67 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.90iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - 1.17T + T^{2} \)
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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

−9.783204217174190678179498505454, −8.920745572944536769454331651417, −8.222828990679922295863047632285, −7.85121162427367982649615292849, −7.01303886937196917946487237889, −5.98265105597529653249688540087, −4.80545692100670486715007249162, −3.69816216846784578691000396805, −2.98315554923513633769343270466, −2.38620615813914264919516822009, 0.846690905699067396259538900009, 1.41952920751712929483755009584, 2.94057577362820738801121345018, 3.67415020702712194842293543484, 4.54897024846794487396271602195, 6.26140007891167102240584126979, 7.30350449979723696396478184827, 7.70522755585354991027790843996, 8.099334849553862467197069493281, 8.785596860210251539263363391943

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