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Properties

Label 1-103-103.49-r0-0-0
Degree $1$
Conductor $103$
Sign $0.512 + 0.858i$
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  + (−0.952 + 0.303i)2-s + (0.932 + 0.361i)3-s + (0.816 − 0.577i)4-s + (0.881 + 0.473i)5-s + (−0.998 − 0.0615i)6-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.969 − 0.243i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.650 + 0.759i)15-s + (0.332 − 0.943i)16-s + (−0.998 + 0.0615i)17-s + ⋯
L(s)  = 1  + (−0.952 + 0.303i)2-s + (0.932 + 0.361i)3-s + (0.816 − 0.577i)4-s + (0.881 + 0.473i)5-s + (−0.998 − 0.0615i)6-s + (−0.389 + 0.920i)7-s + (−0.602 + 0.798i)8-s + (0.739 + 0.673i)9-s + (−0.982 − 0.183i)10-s + (0.213 − 0.976i)11-s + (0.969 − 0.243i)12-s + (−0.602 − 0.798i)13-s + (0.0922 − 0.995i)14-s + (0.650 + 0.759i)15-s + (0.332 − 0.943i)16-s + (−0.998 + 0.0615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8568580756 + 0.4866987342i\)
\(L(\frac12)\) \(\approx\) \(0.8568580756 + 0.4866987342i\)
\(L(1)\) \(\approx\) \(0.9348614656 + 0.3346564479i\)
\(L(1)\) \(\approx\) \(0.9348614656 + 0.3346564479i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.952 + 0.303i)T \)
3 \( 1 + (0.932 + 0.361i)T \)
5 \( 1 + (0.881 + 0.473i)T \)
7 \( 1 + (-0.389 + 0.920i)T \)
11 \( 1 + (0.213 - 0.976i)T \)
13 \( 1 + (-0.602 - 0.798i)T \)
17 \( 1 + (-0.998 + 0.0615i)T \)
19 \( 1 + (-0.153 + 0.988i)T \)
23 \( 1 + (0.739 - 0.673i)T \)
29 \( 1 + (-0.0307 - 0.999i)T \)
31 \( 1 + (-0.982 + 0.183i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (0.881 - 0.473i)T \)
43 \( 1 + (0.969 + 0.243i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.779 - 0.626i)T \)
59 \( 1 + (-0.389 - 0.920i)T \)
61 \( 1 + (0.445 - 0.895i)T \)
67 \( 1 + (0.992 + 0.122i)T \)
71 \( 1 + (-0.0307 + 0.999i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (-0.850 + 0.526i)T \)
83 \( 1 + (0.992 - 0.122i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (-0.998 - 0.0615i)T \)
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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

−29.44518487083890114999682764833, −28.84964527681054606706073248408, −27.48014249857023030481441169216, −26.26612583536659938909110476787, −25.82330263666103942149527291733, −24.79002636598930714899357673654, −23.87606319802558606131574128317, −21.94497392623321131000660719581, −20.873431246100236552626989658364, −19.99571831267212465501570423854, −19.418371366550517639108330738457, −17.91544194085999972565357885842, −17.27782615201159796416655723037, −16.03078254832442514239567338538, −14.59864377890027541080092997546, −13.29512745773879664533272566651, −12.53182888907675062068076064421, −10.79506163221499231399977375770, −9.36624228647225966088188529717, −9.197485945631902935623135854806, −7.38596836188031218215415195308, −6.74422007574480821789271363095, −4.292465691967915789005871682201, −2.57216126304016297630795555766, −1.45890412321110666021540257782, 2.08803622589980571074910332845, 3.0469782472007536689748831374, 5.49164933258096785318634957587, 6.63092843479735715091077976472, 8.161305351147716370079858430429, 9.0888648181904712154413287927, 9.95543163486207851974383941423, 11.0102668560189264528950484079, 12.86920315416678853312553893124, 14.28946145167539500518609334026, 15.09377042472246439530627319627, 16.11963049911059674076368522864, 17.35519666970323461237971349152, 18.61886358892113071415765748234, 19.17716020061967527540782127857, 20.43750033130736857033171016849, 21.47439250419123861184151843409, 22.42665529132454877926373864575, 24.64799733065175002043445893666, 24.87413148167139239502798548202, 25.93168734905950734403670730714, 26.74886632118809078480972634607, 27.6107581658251707682183091615, 28.98496739538546346957437242746, 29.65768768633363072085933342288

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