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Properties

Label 1-2340-2340.47-r0-0-0
Degree $1$
Conductor $2340$
Sign $-0.905 + 0.424i$
Analytic cond. $10.8669$
Root an. cond. $10.8669$
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  + (0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + i·17-s − i·19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + 37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + i·53-s + (−0.866 − 0.5i)59-s + (−0.5 − 0.866i)61-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.866 + 0.5i)11-s + i·17-s − i·19-s + (−0.866 − 0.5i)23-s + (−0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + 37-s + (0.866 + 0.5i)41-s + (−0.866 + 0.5i)43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)49-s + i·53-s + (−0.866 − 0.5i)59-s + (−0.5 − 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.905 + 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.905 + 0.424i$
Analytic conductor: \(10.8669\)
Root analytic conductor: \(10.8669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (0:\ ),\ -0.905 + 0.424i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1851119471 + 0.8306093460i\)
\(L(\frac12)\) \(\approx\) \(0.1851119471 + 0.8306093460i\)
\(L(1)\) \(\approx\) \(0.8839735949 + 0.2661213965i\)
\(L(1)\) \(\approx\) \(0.8839735949 + 0.2661213965i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + iT \)
19 \( 1 - iT \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + iT \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - iT \)
97 \( 1 + (0.5 + 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.460392516653396035861176193265, −18.30429721652552449228602349334, −18.04236045077257970388340897666, −17.143473655265168236396164409473, −16.39868867904423674225844871492, −15.79861898416116258659277834453, −14.99470284866296393268381558685, −14.08944544096971637350353549060, −13.56292311650711677319681196414, −12.99275699354821554770061477686, −11.87525096544317449781566609909, −11.22368951152021668245576834895, −10.63000543521173052314610034981, −9.80398052565587384801002953324, −9.02735852567861210673664899695, −8.02050188286179195102234731918, −7.54673885918080600259816728571, −6.73956774329365573800665445244, −5.73321219830397928359453253141, −4.93297415602182387920098255363, −4.26374179582118006816448369606, −3.22216592521777682372424262942, −2.45380418483377108576545597906, −1.30603219031549237560450915617, −0.272604562850733653481663475588, 1.46885894943458690384577442130, 2.19443946507357258911468199877, 3.02175806909497465820097210909, 4.1812960856101244702280923337, 4.82702660046053061952308972607, 5.872537116012152025990810666445, 6.21134570173056054582447314637, 7.58741837460807614834716753967, 8.076250445060005514976190589489, 8.72887700601593184826160241785, 9.83043312325685566436201582632, 10.286916096293521698886224646919, 11.249298565349805563620861222683, 11.98819803429686915263287324056, 12.63266666136444382434583014040, 13.28655610700709987511040398293, 14.35579403164885171161940134306, 14.86401305398528295819113264449, 15.559553498468918482813415297568, 16.23824345015706867214683393331, 17.14157455571341024461027674580, 17.84517959313130403229187435810, 18.50841022346554241655932368567, 18.98495961977444772148576046327, 20.0240036131470532488892233810

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