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Properties

Label 1-731-731.36-r0-0-0
Degree $1$
Conductor $731$
Sign $-0.575 - 0.817i$
Analytic cond. $3.39474$
Root an. cond. $3.39474$
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  + i·2-s + (0.258 + 0.965i)3-s − 4-s + (−0.258 − 0.965i)5-s + (−0.965 + 0.258i)6-s + (−0.258 + 0.965i)7-s − i·8-s + (−0.866 + 0.5i)9-s + (0.965 − 0.258i)10-s + (0.707 + 0.707i)11-s + (−0.258 − 0.965i)12-s + (0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.866 − 0.5i)15-s + 16-s + ⋯
L(s)  = 1  + i·2-s + (0.258 + 0.965i)3-s − 4-s + (−0.258 − 0.965i)5-s + (−0.965 + 0.258i)6-s + (−0.258 + 0.965i)7-s − i·8-s + (−0.866 + 0.5i)9-s + (0.965 − 0.258i)10-s + (0.707 + 0.707i)11-s + (−0.258 − 0.965i)12-s + (0.5 + 0.866i)13-s + (−0.965 − 0.258i)14-s + (0.866 − 0.5i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(731\)    =    \(17 \cdot 43\)
Sign: $-0.575 - 0.817i$
Analytic conductor: \(3.39474\)
Root analytic conductor: \(3.39474\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{731} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 731,\ (0:\ ),\ -0.575 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3438492493 + 0.6624507805i\)
\(L(\frac12)\) \(\approx\) \(-0.3438492493 + 0.6624507805i\)
\(L(1)\) \(\approx\) \(0.4918868007 + 0.6745248076i\)
\(L(1)\) \(\approx\) \(0.4918868007 + 0.6745248076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
43 \( 1 \)
good2 \( 1 + iT \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
23 \( 1 + (-0.965 + 0.258i)T \)
29 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (0.258 + 0.965i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 - T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 - iT \)
61 \( 1 + (-0.258 + 0.965i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.965 - 0.258i)T \)
73 \( 1 + (0.965 + 0.258i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 + (-0.866 - 0.5i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−22.0920488227319428993942334150, −21.04582396088985065529782351984, −20.05949555295482771700867988885, −19.59724318684022362630454474241, −18.92825161630434217471036237126, −18.16858283476784082354885137075, −17.49554814856898201519609526734, −16.56942409678495255682478190265, −15.0655439012596355066504699898, −14.17682084090804448833592486741, −13.73897173814997180474035754422, −12.89081725753183353492181370927, −12.005937756587506288394507874, −11.15389947493269888305511264456, −10.53325842373010790658605069895, −9.59905030308244918955102199759, −8.271623231113375449155611753793, −7.86738691201275280274368460231, −6.54812550100054651023769732851, −5.96373633513154596447524363375, −4.15891635598120326884389410587, −3.46087168588449029858338842545, −2.66282040108674681705754126131, −1.48520705602469116325648619353, −0.35163959195844792378902878464, 1.73596912622651062164588233682, 3.37277090504201622458401753191, 4.4059194365260397249455483599, 4.837860432710760182568151867343, 5.924577496394069870065407003774, 6.76024278326890280957025094386, 8.25055257535901922107037914860, 8.700299718643103172220874536787, 9.35879234082697919039583915276, 10.10059294747300151203947913068, 11.628982991278923836451950247835, 12.33589736046351378290501498019, 13.383526643855347418177247474741, 14.293897002230672990237617656716, 15.084351584058608490031891327819, 15.82297641816103209403092168285, 16.25054561723418613380560058859, 17.129031200509264499105156859617, 17.851407110068185124687050098545, 19.135911104010022515234081020913, 19.7009828306519689957465913036, 20.80158407141873614606040684198, 21.64462882266726101861756403326, 22.13552605542997457714878664611, 23.17243875618644727351089374097

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