Properties

Label 1-103-103.15-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: $\chi_{103} (15, \cdot )$
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

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

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