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Properties

Label 1-103-103.4-r0-0-0
Degree $1$
Conductor $103$
Sign $0.0762 - 0.997i$
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  + (0.969 − 0.243i)2-s + (−0.602 − 0.798i)3-s + (0.881 − 0.473i)4-s + (0.650 − 0.759i)5-s + (−0.779 − 0.626i)6-s + (−0.952 + 0.303i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (−0.696 − 0.717i)11-s + (−0.908 − 0.417i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (−0.998 − 0.0615i)15-s + (0.552 − 0.833i)16-s + (−0.779 + 0.626i)17-s + ⋯
L(s)  = 1  + (0.969 − 0.243i)2-s + (−0.602 − 0.798i)3-s + (0.881 − 0.473i)4-s + (0.650 − 0.759i)5-s + (−0.779 − 0.626i)6-s + (−0.952 + 0.303i)7-s + (0.739 − 0.673i)8-s + (−0.273 + 0.961i)9-s + (0.445 − 0.895i)10-s + (−0.696 − 0.717i)11-s + (−0.908 − 0.417i)12-s + (0.739 + 0.673i)13-s + (−0.850 + 0.526i)14-s + (−0.998 − 0.0615i)15-s + (0.552 − 0.833i)16-s + (−0.779 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0762 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(103\)
Sign: $0.0762 - 0.997i$
Analytic conductor: \(0.478329\)
Root analytic conductor: \(0.478329\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{103} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 103,\ (0:\ ),\ 0.0762 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.094912957 - 1.014387404i\)
\(L(\frac12)\) \(\approx\) \(1.094912957 - 1.014387404i\)
\(L(1)\) \(\approx\) \(1.299404314 - 0.7077164490i\)
\(L(1)\) \(\approx\) \(1.299404314 - 0.7077164490i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad103 \( 1 \)
good2 \( 1 + (0.969 - 0.243i)T \)
3 \( 1 + (-0.602 - 0.798i)T \)
5 \( 1 + (0.650 - 0.759i)T \)
7 \( 1 + (-0.952 + 0.303i)T \)
11 \( 1 + (-0.696 - 0.717i)T \)
13 \( 1 + (0.739 + 0.673i)T \)
17 \( 1 + (-0.779 + 0.626i)T \)
19 \( 1 + (0.992 + 0.122i)T \)
23 \( 1 + (-0.273 - 0.961i)T \)
29 \( 1 + (0.332 + 0.943i)T \)
31 \( 1 + (0.445 + 0.895i)T \)
37 \( 1 + (0.0922 + 0.995i)T \)
41 \( 1 + (0.650 + 0.759i)T \)
43 \( 1 + (-0.908 + 0.417i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.389 - 0.920i)T \)
59 \( 1 + (-0.952 - 0.303i)T \)
61 \( 1 + (0.932 + 0.361i)T \)
67 \( 1 + (0.213 + 0.976i)T \)
71 \( 1 + (0.332 - 0.943i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.982 - 0.183i)T \)
83 \( 1 + (0.213 - 0.976i)T \)
89 \( 1 + (-0.850 + 0.526i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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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

−29.9769261403241116729782735327, −29.14857977528421123726420611657, −28.342130205647607071643404722402, −26.5420595153692474471676256047, −26.00875640336517909932376108890, −24.95008183699450474509095284235, −23.26974331582006515481668199539, −22.77084682648439303354369669669, −22.027086293107105577959563712186, −20.94990283522011022954340388345, −20.04997645356750677038001910895, −18.138574233940104719570667928868, −17.18683376117786239300235053429, −15.76129643343150440359552894745, −15.44836060513686860640115985995, −13.8897688601927961585874679411, −13.03380472715662069885176327443, −11.587152951064709391988036603482, −10.52548638735420273145624969641, −9.61536460425759697577687081303, −7.34493168991064320042857996238, −6.22599325090329675402325427553, −5.33538681431073794451052783390, −3.83430880172461498087069194044, −2.71196285707853449338202300691, 1.4193788654711751177762254474, 2.90224027654912066057590551734, 4.80665532566091353474289173776, 5.96571169219864111646928094099, 6.60708401360711659976906842300, 8.47956196154682661144246605862, 10.16195273170005261371044186495, 11.395463121899873353321349145215, 12.57898875185432901792058622812, 13.19958271796612836740775216413, 14.02257137736413445188286136412, 16.03652342467481736786729709302, 16.43153491175612670253525311795, 18.079414511298976631066819654491, 19.14345597576279349599826041228, 20.23503157682466941620945915686, 21.50573523794697109529669079909, 22.25753868589715580035934397228, 23.429833264223477847455988476783, 24.21709764257427311982253466598, 25.02291934825425483851600809365, 26.09616925933666688481396801054, 28.31560932980921957985069039545, 28.813940029211249889978419532387, 29.331286508759574265961697586367

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