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Properties

Label 1-103-103.36-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: $\chi_{103} (36, \cdot )$
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

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

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