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Properties

Label 1-103-103.72-r0-0-0
Degree $1$
Conductor $103$
Sign $0.139 + 0.990i$
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  + (−0.850 − 0.526i)2-s + (−0.273 + 0.961i)3-s + (0.445 + 0.895i)4-s + (0.932 + 0.361i)5-s + (0.739 − 0.673i)6-s + (0.0922 + 0.995i)7-s + (0.0922 − 0.995i)8-s + (−0.850 − 0.526i)9-s + (−0.602 − 0.798i)10-s + (−0.850 − 0.526i)11-s + (−0.982 + 0.183i)12-s + (0.0922 + 0.995i)13-s + (0.445 − 0.895i)14-s + (−0.602 + 0.798i)15-s + (−0.602 + 0.798i)16-s + (0.739 + 0.673i)17-s + ⋯
L(s)  = 1  + (−0.850 − 0.526i)2-s + (−0.273 + 0.961i)3-s + (0.445 + 0.895i)4-s + (0.932 + 0.361i)5-s + (0.739 − 0.673i)6-s + (0.0922 + 0.995i)7-s + (0.0922 − 0.995i)8-s + (−0.850 − 0.526i)9-s + (−0.602 − 0.798i)10-s + (−0.850 − 0.526i)11-s + (−0.982 + 0.183i)12-s + (0.0922 + 0.995i)13-s + (0.445 − 0.895i)14-s + (−0.602 + 0.798i)15-s + (−0.602 + 0.798i)16-s + (0.739 + 0.673i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.139 + 0.990i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (72, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.139 + 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5000246879 + 0.4344388123i\)
\(L(\frac12)\) \(\approx\) \(0.5000246879 + 0.4344388123i\)
\(L(1)\) \(\approx\) \(0.6757216355 + 0.2421131578i\)
\(L(1)\) \(\approx\) \(0.6757216355 + 0.2421131578i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.850 - 0.526i)T \)
3 \( 1 + (-0.273 + 0.961i)T \)
5 \( 1 + (0.932 + 0.361i)T \)
7 \( 1 + (0.0922 + 0.995i)T \)
11 \( 1 + (-0.850 - 0.526i)T \)
13 \( 1 + (0.0922 + 0.995i)T \)
17 \( 1 + (0.739 + 0.673i)T \)
19 \( 1 + (-0.273 - 0.961i)T \)
23 \( 1 + (-0.850 + 0.526i)T \)
29 \( 1 + (0.932 + 0.361i)T \)
31 \( 1 + (-0.602 + 0.798i)T \)
37 \( 1 + (-0.982 + 0.183i)T \)
41 \( 1 + (0.932 - 0.361i)T \)
43 \( 1 + (-0.982 - 0.183i)T \)
47 \( 1 + T \)
53 \( 1 + (-0.273 - 0.961i)T \)
59 \( 1 + (0.0922 - 0.995i)T \)
61 \( 1 + (0.739 + 0.673i)T \)
67 \( 1 + (0.0922 - 0.995i)T \)
71 \( 1 + (0.932 - 0.361i)T \)
73 \( 1 + (0.932 - 0.361i)T \)
79 \( 1 + (0.932 + 0.361i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (0.445 - 0.895i)T \)
97 \( 1 + (0.739 - 0.673i)T \)
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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

−29.53503740939200578340895973938, −28.57110205987427478849039863000, −27.62661157909586211378825634712, −26.230375624621335237418517377799, −25.3740776387374227958280788196, −24.65776560475219536437365694556, −23.54524801688615855246863525312, −22.81637092399494318518468977124, −20.72594820657424707748535495316, −20.09849240344835617333757143278, −18.63450390205070934215418660312, −17.87627246567599366184478305411, −17.11787702213067551703923707877, −16.1798311468833426296340768009, −14.44474355027602949221472862428, −13.544417234576292479822035885204, −12.35836446493925517877364528435, −10.64037558481902815814245293273, −9.93105044922241600391897207072, −8.218787287212564648175594366495, −7.45807937356234645474327190993, −6.16575025298891133906723053740, −5.18671922993761880088582916769, −2.30662257814806333644841793899, −0.92566925149760061401325964392, 2.11863990405197156579749173703, 3.34916491441141712035829908734, 5.24814638420250155508287741664, 6.48539398749920077978338335088, 8.45778058271269750360832439956, 9.33006653629820442873106814040, 10.32851012275833067948425326843, 11.21894025125852262548357902324, 12.41467375273783916420563377944, 14.01288651844410184303698246925, 15.44922888721154062536630448854, 16.3756066339590516852856172981, 17.5063976664786233593105146943, 18.33986332864057426021939017750, 19.44155688003279699933745761261, 21.044422849244129669329225270029, 21.47937320379319778317655637728, 22.10805696641502667239838388773, 23.89037801547333790433441666868, 25.53355175411829979670041485467, 25.9887130868306135642451005773, 26.998520812311856948490124876157, 28.238300194125783067132323879492, 28.654241005067777890984038595232, 29.66771307039899092659811206827

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