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Properties

Label 1-103-103.30-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.451 + 0.892i$
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  + (0.0922 + 0.995i)2-s + (0.739 + 0.673i)3-s + (−0.982 + 0.183i)4-s + (0.445 − 0.895i)5-s + (−0.602 + 0.798i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (0.0922 + 0.995i)9-s + (0.932 + 0.361i)10-s + (0.0922 + 0.995i)11-s + (−0.850 − 0.526i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (0.932 − 0.361i)15-s + (0.932 − 0.361i)16-s + (−0.602 − 0.798i)17-s + ⋯
L(s)  = 1  + (0.0922 + 0.995i)2-s + (0.739 + 0.673i)3-s + (−0.982 + 0.183i)4-s + (0.445 − 0.895i)5-s + (−0.602 + 0.798i)6-s + (−0.273 + 0.961i)7-s + (−0.273 − 0.961i)8-s + (0.0922 + 0.995i)9-s + (0.932 + 0.361i)10-s + (0.0922 + 0.995i)11-s + (−0.850 − 0.526i)12-s + (−0.273 + 0.961i)13-s + (−0.982 − 0.183i)14-s + (0.932 − 0.361i)15-s + (0.932 − 0.361i)16-s + (−0.602 − 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.451 + 0.892i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.451 + 0.892i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6414095373 + 1.043757616i\)
\(L(\frac12)\) \(\approx\) \(0.6414095373 + 1.043757616i\)
\(L(1)\) \(\approx\) \(0.9290936898 + 0.8062977315i\)
\(L(1)\) \(\approx\) \(0.9290936898 + 0.8062977315i\)

Euler product

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

−29.61771105874748412147958114790, −29.05450311624981119015408701457, −27.15531505094872791824007374079, −26.58135225263296552668352929152, −25.56483115149262440061295580576, −24.1621599628121149359163197275, −23.091356866538952243998563129830, −22.089999610862704749780248750719, −20.971775968540140297201509165094, −19.85395021228238067113792033279, −19.222159877044717006215366781876, −18.13993806205810113079924119398, −17.28054535611644199711899121634, −15.15125165294285866683585442793, −13.82615338599251623249253323713, −13.61804117351521481132297144099, −12.24274666849762719924006853803, −10.79213281470230625163242473411, −9.992122847528716510423518344217, −8.59225553753054881208290670673, −7.30398195085041592406396758454, −5.85068177795455877368550765301, −3.65732091122138563384018258622, −2.927364418974554271016173579353, −1.32898461908712002210946281177, 2.42330579546674959258965043312, 4.41537065914587702624824559298, 5.10001762385334328491275422111, 6.68787038580889016350932841399, 8.22337162616655734280428043552, 9.26927253562428477413383036333, 9.668092698827119936418462684364, 12.0638563389911065153218577289, 13.27698589161444175638664880652, 14.23472116515258916382400978026, 15.41289016328034244740409922991, 16.06878428423959331394708545771, 17.170144095031605040506833286822, 18.38939697594308983854963652858, 19.71179250337833967559014696157, 20.95231546204836080225265976128, 21.85225615991728571193783265597, 22.80300664444085569652790202301, 24.58736469834459100709076172621, 24.801043330844296726424318352938, 25.93905213302362669275999899559, 26.72951476885820626008601961795, 28.083159968338702365746393690311, 28.5689217178733066185028031164, 30.72327223756490043193038263904

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