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Properties

Label 1-2340-2340.2207-r1-0-0
Degree $1$
Conductor $2340$
Sign $-0.156 - 0.987i$
Analytic cond. $251.467$
Root an. cond. $251.467$
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  + (−0.866 − 0.5i)7-s − 11-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s + i·53-s + 59-s + (−0.5 + 0.866i)61-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)7-s − 11-s + (−0.866 + 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s + 29-s + (−0.5 + 0.866i)31-s + (0.866 + 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + (−0.866 + 0.5i)47-s + (0.5 + 0.866i)49-s + i·53-s + 59-s + (−0.5 + 0.866i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.156 - 0.987i$
Analytic conductor: \(251.467\)
Root analytic conductor: \(251.467\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2340,\ (1:\ ),\ -0.156 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2592964273 - 0.3036897659i\)
\(L(\frac12)\) \(\approx\) \(0.2592964273 - 0.3036897659i\)
\(L(1)\) \(\approx\) \(0.7596790568 + 0.03675501819i\)
\(L(1)\) \(\approx\) \(0.7596790568 + 0.03675501819i\)

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 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 - T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + iT \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (0.866 - 0.5i)T \)
71 \( 1 + (0.5 + 0.866i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.83491078037555094037754934524, −18.78834928691669836522651717317, −18.1694170679162720946302640692, −17.73406198711066723865631928537, −16.45527771693059232900904552414, −16.10866689173471719416812897857, −15.4001146688547983950147248820, −14.73043694264663629638119893553, −13.563424909450418744269992723933, −13.235579864047787302445299057317, −12.449889112847883543063782149399, −11.60555238892956893737837356898, −10.93535281760840482339797887136, −9.91197921344544906892131641344, −9.52300171324074136116095092517, −8.53025864884826447372658485122, −7.883873653785683775562206653371, −6.82474143246536830653431470643, −6.31649958481377283507484866250, −5.30861685101433184459529779069, −4.67436490104500408944809761378, −3.56070095374904389232799629705, −2.69103822549143022618370819202, −2.14486305679988557860469909526, −0.58794312959260482236302525494, 0.10514347241913421290773143694, 1.29403383752625307143597730268, 2.35064952900625751734567222604, 3.25417099461532912007377614536, 3.96497039626314906665258291545, 4.92426652558153444676682354088, 5.81587620559321411109577745429, 6.538517027871054210460737364257, 7.35108166121445159216770628539, 8.1120694686564371165971044751, 8.884555602822401989666934461258, 9.99402982127473559288212442814, 10.22283459708314863544321400942, 11.12661648864306097989489375291, 12.08383521058637903538466988033, 12.76059985437755792628951647038, 13.46134166048068834256715713851, 14.00672275477224499041044025975, 14.99479678140612356648397909397, 15.881831540554527259172422812006, 16.13454501120476620661640988589, 17.087015921277177592436606143451, 17.831065858231688110153674550025, 18.50849048985558195178317780099, 19.24193963165294043414084055802

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