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Properties

Label 1-113-113.61-r0-0-0
Degree $1$
Conductor $113$
Sign $0.294 - 0.955i$
Analytic cond. $0.524769$
Root an. cond. $0.524769$
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  + (0.900 − 0.433i)2-s + (0.846 − 0.532i)3-s + (0.623 − 0.781i)4-s + (−0.846 − 0.532i)5-s + (0.532 − 0.846i)6-s + (−0.222 + 0.974i)7-s + (0.222 − 0.974i)8-s + (0.433 − 0.900i)9-s + (−0.993 − 0.111i)10-s + (−0.781 + 0.623i)11-s + (0.111 − 0.993i)12-s + (0.974 + 0.222i)13-s + (0.222 + 0.974i)14-s − 15-s + (−0.222 − 0.974i)16-s + (−0.943 − 0.330i)17-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.846 − 0.532i)3-s + (0.623 − 0.781i)4-s + (−0.846 − 0.532i)5-s + (0.532 − 0.846i)6-s + (−0.222 + 0.974i)7-s + (0.222 − 0.974i)8-s + (0.433 − 0.900i)9-s + (−0.993 − 0.111i)10-s + (−0.781 + 0.623i)11-s + (0.111 − 0.993i)12-s + (0.974 + 0.222i)13-s + (0.222 + 0.974i)14-s − 15-s + (−0.222 − 0.974i)16-s + (−0.943 − 0.330i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.294 - 0.955i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.294 - 0.955i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.511707683 - 1.116050036i\)
\(L(\frac12)\) \(\approx\) \(1.511707683 - 1.116050036i\)
\(L(1)\) \(\approx\) \(1.611864444 - 0.7827682471i\)
\(L(1)\) \(\approx\) \(1.611864444 - 0.7827682471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad113 \( 1 \)
good2 \( 1 + (0.900 - 0.433i)T \)
3 \( 1 + (0.846 - 0.532i)T \)
5 \( 1 + (-0.846 - 0.532i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
11 \( 1 + (-0.781 + 0.623i)T \)
13 \( 1 + (0.974 + 0.222i)T \)
17 \( 1 + (-0.943 - 0.330i)T \)
19 \( 1 + (0.532 + 0.846i)T \)
23 \( 1 + (-0.532 + 0.846i)T \)
29 \( 1 + (0.943 + 0.330i)T \)
31 \( 1 + (-0.974 - 0.222i)T \)
37 \( 1 + (0.993 + 0.111i)T \)
41 \( 1 + (-0.781 - 0.623i)T \)
43 \( 1 + (0.330 - 0.943i)T \)
47 \( 1 + (0.111 + 0.993i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (-0.532 - 0.846i)T \)
61 \( 1 + (0.781 - 0.623i)T \)
67 \( 1 + (-0.111 + 0.993i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.707 + 0.707i)T \)
79 \( 1 + (-0.993 + 0.111i)T \)
83 \( 1 + (-0.900 + 0.433i)T \)
89 \( 1 + (-0.330 - 0.943i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−30.20771338444463441681271502294, −28.66539171965070474995619561931, −26.952306922085208147277001005953, −26.442775214161552681679843958078, −25.69446289116629189511953116707, −24.261452750426964860939169334262, −23.473019675493777012645318383486, −22.44931779388682893524395999889, −21.46064229176478990224886749924, −20.28351860565863497112024983144, −19.724453805588692812056257376257, −18.16337887653743201912383963417, −16.3393495999812259915731895322, −15.837869060632016041002661094323, −14.8410640211883314491028383255, −13.74660136767057797135523701576, −13.076653547062759025953909329357, −11.23109399872474560564184874670, −10.52401204827705389088672880321, −8.50838604532414536912766307026, −7.677999067888902323314675119827, −6.49868828420890300825186606414, −4.64664011951541139013715954267, −3.7046608917975419179171485411, −2.77922858734708810303487635318, 1.71782006758190474231162670815, 3.03641759551238842171180071558, 4.212823249086902425937344256259, 5.69183621734883906475560873828, 7.15776563750538245910779705395, 8.4186856707592477962379246908, 9.61489425451469073046936044099, 11.360239869545112086231668567261, 12.36961585847596123191794304885, 13.046214303522159540373061380084, 14.20654391047094319533166560659, 15.59187822009674776148540817807, 15.76358758894604164210081454670, 18.24850077399136026531906415585, 18.99130238787649231980557603399, 20.15781428364994250282948803177, 20.64881851727550538691496198739, 21.89709439492458193024257761759, 23.25687018138749625569071594241, 23.89633307337782950675892861635, 24.95522285691956996102132323046, 25.716725727050282989340010034754, 27.30058282858857388055940182992, 28.49580333160493804633301629755, 29.15509660978582522549397314478

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