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Properties

Label 1-103-103.55-r0-0-0
Degree $1$
Conductor $103$
Sign $-0.503 + 0.864i$
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  + (−0.0307 − 0.999i)2-s + (−0.273 − 0.961i)3-s + (−0.998 + 0.0615i)4-s + (−0.779 − 0.626i)5-s + (−0.952 + 0.303i)6-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (0.881 + 0.473i)11-s + (0.332 + 0.943i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (−0.389 + 0.920i)15-s + (0.992 − 0.122i)16-s + (−0.952 − 0.303i)17-s + ⋯
L(s)  = 1  + (−0.0307 − 0.999i)2-s + (−0.273 − 0.961i)3-s + (−0.998 + 0.0615i)4-s + (−0.779 − 0.626i)5-s + (−0.952 + 0.303i)6-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (−0.602 + 0.798i)10-s + (0.881 + 0.473i)11-s + (0.332 + 0.943i)12-s + (0.0922 − 0.995i)13-s + (0.445 + 0.895i)14-s + (−0.389 + 0.920i)15-s + (0.992 − 0.122i)16-s + (−0.952 − 0.303i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.503 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $-0.503 + 0.864i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ -0.503 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1662143450 - 0.2890874807i\)
\(L(\frac12)\) \(\approx\) \(-0.1662143450 - 0.2890874807i\)
\(L(1)\) \(\approx\) \(0.3103024611 - 0.4581160712i\)
\(L(1)\) \(\approx\) \(0.3103024611 - 0.4581160712i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.0307 - 0.999i)T \)
3 \( 1 + (-0.273 - 0.961i)T \)
5 \( 1 + (-0.779 - 0.626i)T \)
7 \( 1 + (-0.908 + 0.417i)T \)
11 \( 1 + (0.881 + 0.473i)T \)
13 \( 1 + (0.0922 - 0.995i)T \)
17 \( 1 + (-0.952 - 0.303i)T \)
19 \( 1 + (-0.696 - 0.717i)T \)
23 \( 1 + (-0.850 - 0.526i)T \)
29 \( 1 + (-0.153 + 0.988i)T \)
31 \( 1 + (-0.602 - 0.798i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (-0.779 + 0.626i)T \)
43 \( 1 + (0.332 - 0.943i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.969 - 0.243i)T \)
59 \( 1 + (-0.908 - 0.417i)T \)
61 \( 1 + (0.739 - 0.673i)T \)
67 \( 1 + (0.816 - 0.577i)T \)
71 \( 1 + (-0.153 - 0.988i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (0.932 - 0.361i)T \)
83 \( 1 + (0.816 + 0.577i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (-0.952 + 0.303i)T \)
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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

−30.84724013297100352722652119680, −29.32006774389938850723529824737, −28.014705803489395097637307450469, −27.1266943912518210042239014164, −26.39420176133456611356060773864, −25.723322729484962319289918801606, −24.12628612025326110815107026499, −23.13793474095967746212046441373, −22.42313378795977610830649748118, −21.600374289025509346930985483242, −19.75902938978984401103663866416, −18.96425468618069342335038077925, −17.419960774300668147317721864711, −16.43751340827863383281588543094, −15.78961331958518640378079499953, −14.724133173049249052452041212535, −13.79064380320245852361728624729, −12.038705175947073449163993352395, −10.71869757902833254056470692197, −9.55315405931599509103811199785, −8.48366387449469125391373003054, −6.84682649127055516444250026467, −6.097118700725824490678191044129, −4.20456677497265157850811520865, −3.671065271523040472086373591140, 0.34449873967682198370999898058, 2.11687291802118300338654106015, 3.63806948784974350793461372934, 5.16327024846857587602581458483, 6.75183371414977345025836466469, 8.2927024431897755950654533614, 9.209731694978536387806930952070, 10.85311526914942341875622953269, 12.01856809164057262291986403017, 12.6230345833290563320846349305, 13.47028213255903450943763714212, 15.12616873881379699365629123312, 16.669384859538956190121297196960, 17.76425929302294328532865038403, 18.801178378175310456931225381762, 19.89458338843097828161119908767, 20.10805149635400625066932600894, 22.12459194899692413027226928127, 22.69931540888896091803766322715, 23.75602795715826057535406117537, 24.86812295382161518123289314357, 26.05669326660232457975802129010, 27.6726202894156158264868522628, 28.14182252072706318118481357546, 29.16841488706249884609411620944

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