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Properties

Label 1-1033-1033.14-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.487 - 0.873i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − 6-s − 7-s + 8-s + (−0.5 − 0.866i)9-s + (0.866 + 0.5i)10-s + (−0.866 + 0.5i)11-s + (0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + (0.5 + 0.866i)14-s + i·15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4869325224 - 0.2858816928i\)
\(L(\frac12)\) \(\approx\) \(0.4869325224 - 0.2858816928i\)
\(L(1)\) \(\approx\) \(0.5229776977 - 0.2873992747i\)
\(L(1)\) \(\approx\) \(0.5229776977 - 0.2873992747i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.866 + 0.5i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 - iT \)
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

−21.89140669753379497189170635501, −20.78089266297614232055413510522, −19.82432594632350068879615881126, −19.42749573421266709025424148707, −18.84565749782504909940177364474, −17.504567625258578802660248807595, −16.736945579159782227687776856992, −16.162680440332259076133054985540, −15.36561558183382745851648243699, −15.205457351130866669846499962302, −13.98045355748046504873830923382, −13.18425937393878171274210544055, −12.319926408299695636101235993384, −10.88727299251846254862439889418, −10.3350718610176972191545185012, −9.437408442198227748324379309541, −8.80037176547289187938568057343, −7.89334871717652247805656301447, −7.450842463560886877121885110679, −6.02718487853224319556911058808, −5.29810896184974097449721031283, −4.357593599183114274481409187856, −3.53551883668072720712510540978, −2.40928507163840258869097939468, −0.4849457863207029661498973204, 0.589414544676651740249038846872, 2.18737087753789243239133116662, 2.70825876620611776696518682367, 3.58857935349576709417573715672, 4.49171544178046479150140145761, 6.09754233174521329839689443058, 7.238532982940561253328254092997, 7.59929392848729198371595214815, 8.42341347577179676185249066312, 9.57577704898447558675750231883, 10.022369836493562521826467366999, 11.186919855432724719286444765552, 12.05980136109304000355518989728, 12.514124187639471118958243972942, 13.232529216482497412865050641608, 14.23168446704565060733992354387, 14.98904809318278663569189122020, 16.17429512735202500114506451248, 16.7848513100275973771773872473, 18.05226450668646496110924632922, 18.56729369924020267137303227229, 19.05839377270023514142993913188, 19.81021681846841081656976904505, 20.310939923533390349205704374100, 21.189321452518534318174643317764

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