Properties

Label 1-103-103.83-r0-0-0
Degree $1$
Conductor $103$
Sign $0.911 - 0.411i$
Analytic cond. $0.478329$
Root an. cond. $0.478329$
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  + (−0.779 + 0.626i)2-s + (−0.982 − 0.183i)3-s + (0.213 − 0.976i)4-s + (−0.696 + 0.717i)5-s + (0.881 − 0.473i)6-s + (−0.998 + 0.0615i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (−0.153 − 0.988i)11-s + (−0.389 + 0.920i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.816 − 0.577i)15-s + (−0.908 − 0.417i)16-s + (0.881 + 0.473i)17-s + ⋯
L(s)  = 1  + (−0.779 + 0.626i)2-s + (−0.982 − 0.183i)3-s + (0.213 − 0.976i)4-s + (−0.696 + 0.717i)5-s + (0.881 − 0.473i)6-s + (−0.998 + 0.0615i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (−0.153 − 0.988i)11-s + (−0.389 + 0.920i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (0.816 − 0.577i)15-s + (−0.908 − 0.417i)16-s + (0.881 + 0.473i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.911 - 0.411i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.911 - 0.411i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.911 - 0.411i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3565802418 - 0.07669707686i\)
\(L(\frac12)\) \(\approx\) \(0.3565802418 - 0.07669707686i\)
\(L(1)\) \(\approx\) \(0.4573516698 + 0.04082999598i\)
\(L(1)\) \(\approx\) \(0.4573516698 + 0.04082999598i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.779 + 0.626i)T \)
3 \( 1 + (-0.982 - 0.183i)T \)
5 \( 1 + (-0.696 + 0.717i)T \)
7 \( 1 + (-0.998 + 0.0615i)T \)
11 \( 1 + (-0.153 - 0.988i)T \)
13 \( 1 + (0.445 - 0.895i)T \)
17 \( 1 + (0.881 + 0.473i)T \)
19 \( 1 + (0.332 - 0.943i)T \)
23 \( 1 + (0.932 - 0.361i)T \)
29 \( 1 + (0.969 + 0.243i)T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + (-0.696 - 0.717i)T \)
43 \( 1 + (-0.389 - 0.920i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.650 + 0.759i)T \)
59 \( 1 + (-0.998 - 0.0615i)T \)
61 \( 1 + (-0.850 + 0.526i)T \)
67 \( 1 + (0.552 - 0.833i)T \)
71 \( 1 + (0.969 - 0.243i)T \)
73 \( 1 + (-0.273 + 0.961i)T \)
79 \( 1 + (-0.273 - 0.961i)T \)
83 \( 1 + (0.552 + 0.833i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (0.881 - 0.473i)T \)
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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

−29.34369422528681402130264997813, −28.789039620859280155017241512979, −27.93621145530107578470259120112, −27.18314764391382631307833141485, −26.013116253704801468039096191831, −24.86011856088303577116328825507, −23.29328498879292166210509186355, −22.74264207346881818049939747461, −21.2911347185785599008093672578, −20.48170267584288697752739130332, −19.22967401344560218936059945504, −18.4265754831653313126628231473, −17.01195571408051386967389900814, −16.416196151383008547917825229503, −15.54803963622640511904208011309, −13.13631418954764803520862320063, −12.21395483570423561062263814919, −11.55119959093030193132145341714, −10.0819805853905856459722141668, −9.35869269028445466951099386433, −7.74090127026666411082792851783, −6.57865988147281455717976124331, −4.72600136395866673264155876687, −3.507036392264414284601764422945, −1.24115904735867125897241560696, 0.6351552902398108218456334295, 3.21244792275861692503089156935, 5.33833467418915741631573878132, 6.41596210834789877340695596705, 7.26732733075015764704858115184, 8.613475125440657652878343664319, 10.3227590114594297339881085624, 10.85206551573959938648780539484, 12.2132581113862880814678708557, 13.69720675862537572445715573627, 15.35622403062531061055388099667, 15.98347362006733860357214465581, 16.96190394643528773150234603886, 18.20059400389906322097970494714, 18.93080127193078668357138039516, 19.75130731973612456834189361854, 21.703084536636574984387744032282, 22.945642364014160609661536517347, 23.36352010416005933219170722210, 24.5792487530774960843159697671, 25.7452787414391645349279744844, 26.77453675961760987375477643150, 27.56932921190972778633418919396, 28.58949973332966042349308724123, 29.45275769434806869642941694396

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