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Properties

Label 1-1033-1033.123-r0-0-0
Degree $1$
Conductor $1033$
Sign $0.841 - 0.539i$
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  + (−0.791 + 0.611i)2-s + (0.989 − 0.145i)3-s + (0.252 − 0.967i)4-s + (−0.905 − 0.424i)5-s + (−0.694 + 0.719i)6-s + (−0.791 − 0.611i)7-s + (0.391 + 0.920i)8-s + (0.957 − 0.288i)9-s + (0.976 − 0.217i)10-s + (0.872 − 0.489i)11-s + (0.109 − 0.994i)12-s + (−0.957 + 0.288i)13-s + 14-s + (−0.957 − 0.288i)15-s + (−0.872 − 0.489i)16-s + (0.744 + 0.667i)17-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.989 − 0.145i)3-s + (0.252 − 0.967i)4-s + (−0.905 − 0.424i)5-s + (−0.694 + 0.719i)6-s + (−0.791 − 0.611i)7-s + (0.391 + 0.920i)8-s + (0.957 − 0.288i)9-s + (0.976 − 0.217i)10-s + (0.872 − 0.489i)11-s + (0.109 − 0.994i)12-s + (−0.957 + 0.288i)13-s + 14-s + (−0.957 − 0.288i)15-s + (−0.872 − 0.489i)16-s + (0.744 + 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.841 - 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $0.841 - 0.539i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ 0.841 - 0.539i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086860675 - 0.3184341619i\)
\(L(\frac12)\) \(\approx\) \(1.086860675 - 0.3184341619i\)
\(L(1)\) \(\approx\) \(0.8824878732 + 0.02000058681i\)
\(L(1)\) \(\approx\) \(0.8824878732 + 0.02000058681i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
3 \( 1 + (0.989 - 0.145i)T \)
5 \( 1 + (-0.905 - 0.424i)T \)
7 \( 1 + (-0.791 - 0.611i)T \)
11 \( 1 + (0.872 - 0.489i)T \)
13 \( 1 + (-0.957 + 0.288i)T \)
17 \( 1 + (0.744 + 0.667i)T \)
19 \( 1 + (-0.181 + 0.983i)T \)
23 \( 1 + (0.694 - 0.719i)T \)
29 \( 1 + (0.457 + 0.889i)T \)
31 \( 1 + (0.905 + 0.424i)T \)
37 \( 1 + (-0.872 - 0.489i)T \)
41 \( 1 + (0.957 + 0.288i)T \)
43 \( 1 + (0.520 - 0.853i)T \)
47 \( 1 + (-0.391 - 0.920i)T \)
53 \( 1 + (-0.905 - 0.424i)T \)
59 \( 1 + (0.457 - 0.889i)T \)
61 \( 1 + (-0.457 - 0.889i)T \)
67 \( 1 + (0.581 - 0.813i)T \)
71 \( 1 + (-0.905 + 0.424i)T \)
73 \( 1 + (0.872 - 0.489i)T \)
79 \( 1 + (0.872 - 0.489i)T \)
83 \( 1 + (0.391 - 0.920i)T \)
89 \( 1 + (-0.833 + 0.551i)T \)
97 \( 1 + (-0.833 - 0.551i)T \)
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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

−21.42528190668620783030139090970, −20.69135092418837316088346939485, −19.66889497682793709505728516744, −19.40484481571002530899481892665, −19.00515757267180029042586201395, −17.940185064056149523267603629236, −17.06214081805624993966849363324, −15.99449040578879184139464250607, −15.462442248975102171072690556660, −14.76269326891870160229565686975, −13.67572640695227723884173193564, −12.64609097559283194689164746182, −12.08656611040528470021667766732, −11.3001852021517760818234776333, −10.123891696056143598989053663566, −9.53498041183451027681334005780, −8.95142767070717747775217543205, −7.91710243463273549315941977155, −7.303160477791206411501160084864, −6.577578147376613920566532481530, −4.710997268596525873635506972409, −3.85182661002950292810638644874, −2.819986769396947586799952544439, −2.622210653444421588723731939075, −1.02427704972100614508808398469, 0.69752448225298480602303699604, 1.68812737491162749563691853444, 3.13333731488932032542574487139, 3.92009748117966220729130629121, 4.93055471061995966586005362643, 6.39178749260380395410389790396, 7.01357027344623946276281203997, 7.81052700092576598172168147253, 8.51067994730298792527536411856, 9.2137088524119447896351513184, 10.015376360424607203600238650235, 10.7955065959726092109012165420, 12.15311696543927584134097832770, 12.68147387148419245219778547563, 14.04209626681927159964513854753, 14.438924973452647099993952525086, 15.261586370997558599252330562377, 16.193486539160882761171201149, 16.6337289740561004347687626518, 17.40521141594711826985142532433, 18.83889755337324539455877081169, 19.18143739230700549808889883453, 19.64849373667403546031360460034, 20.34305703617422388935110476184, 21.204203181999049186364545902576

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