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Properties

Label 1-103-103.60-r0-0-0
Degree $1$
Conductor $103$
Sign $0.0861 + 0.996i$
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  + (0.213 + 0.976i)2-s + (0.932 − 0.361i)3-s + (−0.908 + 0.417i)4-s + (−0.0307 + 0.999i)5-s + (0.552 + 0.833i)6-s + (0.992 + 0.122i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (−0.982 + 0.183i)10-s + (−0.952 − 0.303i)11-s + (−0.696 + 0.717i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (0.332 + 0.943i)15-s + (0.650 − 0.759i)16-s + (0.552 − 0.833i)17-s + ⋯
L(s)  = 1  + (0.213 + 0.976i)2-s + (0.932 − 0.361i)3-s + (−0.908 + 0.417i)4-s + (−0.0307 + 0.999i)5-s + (0.552 + 0.833i)6-s + (0.992 + 0.122i)7-s + (−0.602 − 0.798i)8-s + (0.739 − 0.673i)9-s + (−0.982 + 0.183i)10-s + (−0.952 − 0.303i)11-s + (−0.696 + 0.717i)12-s + (−0.602 + 0.798i)13-s + (0.0922 + 0.995i)14-s + (0.332 + 0.943i)15-s + (0.650 − 0.759i)16-s + (0.552 − 0.833i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0861 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.0861 + 0.996i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.0861 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025307017 + 0.9404909195i\)
\(L(\frac12)\) \(\approx\) \(1.025307017 + 0.9404909195i\)
\(L(1)\) \(\approx\) \(1.183324553 + 0.7002281375i\)
\(L(1)\) \(\approx\) \(1.183324553 + 0.7002281375i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.213 + 0.976i)T \)
3 \( 1 + (0.932 - 0.361i)T \)
5 \( 1 + (-0.0307 + 0.999i)T \)
7 \( 1 + (0.992 + 0.122i)T \)
11 \( 1 + (-0.952 - 0.303i)T \)
13 \( 1 + (-0.602 + 0.798i)T \)
17 \( 1 + (0.552 - 0.833i)T \)
19 \( 1 + (-0.779 + 0.626i)T \)
23 \( 1 + (0.739 + 0.673i)T \)
29 \( 1 + (0.881 - 0.473i)T \)
31 \( 1 + (-0.982 - 0.183i)T \)
37 \( 1 + (-0.273 - 0.961i)T \)
41 \( 1 + (-0.0307 - 0.999i)T \)
43 \( 1 + (-0.696 - 0.717i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.153 - 0.988i)T \)
59 \( 1 + (0.992 - 0.122i)T \)
61 \( 1 + (0.445 + 0.895i)T \)
67 \( 1 + (-0.389 + 0.920i)T \)
71 \( 1 + (0.881 + 0.473i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (-0.850 - 0.526i)T \)
83 \( 1 + (-0.389 - 0.920i)T \)
89 \( 1 + (0.0922 + 0.995i)T \)
97 \( 1 + (0.552 + 0.833i)T \)
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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

−29.7216505592317360329111572260, −28.396869622734826920364850200626, −27.6150237166539265027458198806, −26.836767546783987511341775934265, −25.47780328305789779617479878124, −24.2448895316103554271175532492, −23.42603492275696976347466530728, −21.7114505785388442143403025912, −21.04670811605371138460349663953, −20.29763704481455178955547574146, −19.509766383521047621718252475329, −18.17804979567639220289488706706, −17.001946693658152394812554589002, −15.30573725344892673806277178558, −14.49464701705331714171443392710, −13.14826342552255915747580242998, −12.54248506064943789148597206877, −10.8739663757077746128786051776, −9.94707380147559308088128608195, −8.61918613562717689156398500884, −7.963072322244694651281885897495, −5.10769188537599999681886957401, −4.51621594234409596265161903629, −2.89391081364508771844549457207, −1.55642907694174682032742180160, 2.354058953019557096974311067732, 3.78555174668177608568849217555, 5.335191321461767873690410742123, 6.9653762196433403491586786880, 7.66801407567204672014924405311, 8.734785510364582196052141461066, 10.1218538440763744985910435159, 11.834423130578864713010937399193, 13.31039662320661956885180141026, 14.32637161054451524654063777277, 14.789741580731644836442725400, 15.923978726361334020274259077603, 17.50128041111661088365931340700, 18.48708873894265037202092340815, 19.10271461750418872314940396795, 20.99441042749617539441315214914, 21.620240101578327711318038294296, 23.195743985292419241493708442496, 23.91540080149705827710362559692, 25.00354814320790003977976420888, 25.80026411176304547228177930841, 26.81392260964469163572458775598, 27.29767424468082267945283047799, 29.36315645281229252657043634050, 30.399634688133850923043968793818

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