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Properties

Label 1-103-103.59-r0-0-0
Degree $1$
Conductor $103$
Sign $0.764 + 0.644i$
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  + (−0.153 + 0.988i)2-s + (−0.982 + 0.183i)3-s + (−0.952 − 0.303i)4-s + (0.969 − 0.243i)5-s + (−0.0307 − 0.999i)6-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (0.932 − 0.361i)9-s + (0.0922 + 0.995i)10-s + (−0.779 − 0.626i)11-s + (0.992 + 0.122i)12-s + (0.445 + 0.895i)13-s + (0.739 + 0.673i)14-s + (−0.908 + 0.417i)15-s + (0.816 + 0.577i)16-s + (−0.0307 + 0.999i)17-s + ⋯
L(s)  = 1  + (−0.153 + 0.988i)2-s + (−0.982 + 0.183i)3-s + (−0.952 − 0.303i)4-s + (0.969 − 0.243i)5-s + (−0.0307 − 0.999i)6-s + (0.552 − 0.833i)7-s + (0.445 − 0.895i)8-s + (0.932 − 0.361i)9-s + (0.0922 + 0.995i)10-s + (−0.779 − 0.626i)11-s + (0.992 + 0.122i)12-s + (0.445 + 0.895i)13-s + (0.739 + 0.673i)14-s + (−0.908 + 0.417i)15-s + (0.816 + 0.577i)16-s + (−0.0307 + 0.999i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 + 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.764 + 0.644i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.764 + 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7404312503 + 0.2702666084i\)
\(L(\frac12)\) \(\approx\) \(0.7404312503 + 0.2702666084i\)
\(L(1)\) \(\approx\) \(0.7826529369 + 0.2708080028i\)
\(L(1)\) \(\approx\) \(0.7826529369 + 0.2708080028i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (-0.153 + 0.988i)T \)
3 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (0.969 - 0.243i)T \)
7 \( 1 + (0.552 - 0.833i)T \)
11 \( 1 + (-0.779 - 0.626i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
17 \( 1 + (-0.0307 + 0.999i)T \)
19 \( 1 + (0.650 - 0.759i)T \)
23 \( 1 + (0.932 + 0.361i)T \)
29 \( 1 + (-0.696 - 0.717i)T \)
31 \( 1 + (0.0922 - 0.995i)T \)
37 \( 1 + (-0.602 + 0.798i)T \)
41 \( 1 + (0.969 + 0.243i)T \)
43 \( 1 + (0.992 - 0.122i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.332 + 0.943i)T \)
59 \( 1 + (0.552 + 0.833i)T \)
61 \( 1 + (-0.850 - 0.526i)T \)
67 \( 1 + (-0.998 + 0.0615i)T \)
71 \( 1 + (-0.696 + 0.717i)T \)
73 \( 1 + (-0.273 - 0.961i)T \)
79 \( 1 + (-0.273 + 0.961i)T \)
83 \( 1 + (-0.998 - 0.0615i)T \)
89 \( 1 + (0.739 + 0.673i)T \)
97 \( 1 + (-0.0307 - 0.999i)T \)
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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

−29.42299726671044104086972747812, −28.80843082438082989435859134606, −27.93800643479284134175670707409, −27.04607439865187877824630182601, −25.57850132289730352296535175634, −24.53265496371592693555594449791, −22.921572422427114827715115324132, −22.45548401415111403186457463991, −21.20867287037538030201890563921, −20.69297591574056629468462333050, −18.80218608953340130814492089110, −17.992375022046009268760666774522, −17.67240180987779001651295287087, −16.07972200856288381901451595322, −14.457018847409149230004962526226, −13.086482429606886506390748007437, −12.35488898604860467836992279235, −11.0856974575146460992948323728, −10.304384363237977430573528460550, −9.12994260875876941159046403108, −7.503449848369767293886473449140, −5.619226578155656379674509803447, −4.99338125757689895265531658128, −2.78776969561075764017360678571, −1.478515321856110940758922123630, 1.19924211301923770680080413257, 4.232781962571556511406257333509, 5.33672263728583927164800799692, 6.25867232508700236828660478451, 7.46777814572795954499274795980, 8.99943553177065915172894425927, 10.18759844227247031252274379142, 11.18030127660626018558548568752, 13.11408282032089127240482765283, 13.734122996554950408569500370822, 15.19445506329606284775789581128, 16.453261706086624021647189775204, 17.09496518988012439587416784006, 17.883639340490033281067475486362, 18.92267294082599904856656921543, 20.946588458838744947567278018716, 21.6951177779384499285612029439, 22.90249914747136287503856134731, 23.99141161773985035466958809953, 24.34314117247748987717266770491, 26.06339630021526761925723423847, 26.56813761347131612269230023236, 27.89092986845926004468584709479, 28.682802942406451327363617849198, 29.6879678493101268501428453250

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