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Properties

Label 1-113-113.97-r0-0-0
Degree $1$
Conductor $113$
Sign $0.481 + 0.876i$
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  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.222 + 0.974i)10-s + (−0.222 − 0.974i)11-s + (0.222 + 0.974i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)14-s + 15-s + (−0.900 − 0.433i)16-s + (−0.623 − 0.781i)17-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.900 − 0.433i)3-s + (−0.222 + 0.974i)4-s + (0.900 + 0.433i)5-s + (0.900 + 0.433i)6-s + (−0.900 + 0.433i)7-s + (−0.900 + 0.433i)8-s + (0.623 − 0.781i)9-s + (0.222 + 0.974i)10-s + (−0.222 − 0.974i)11-s + (0.222 + 0.974i)12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)14-s + 15-s + (−0.900 − 0.433i)16-s + (−0.623 − 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 113 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(113\)
Sign: $0.481 + 0.876i$
Analytic conductor: \(0.524769\)
Root analytic conductor: \(0.524769\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{113} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 113,\ (0:\ ),\ 0.481 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.543427271 + 0.9135767903i\)
\(L(\frac12)\) \(\approx\) \(1.543427271 + 0.9135767903i\)
\(L(1)\) \(\approx\) \(1.578209804 + 0.6574435116i\)
\(L(1)\) \(\approx\) \(1.578209804 + 0.6574435116i\)

Euler product

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

−29.21966169196061213475846276148, −28.490943982732490343136651056001, −27.242549315212246183737723163879, −26.09120474299457067540010663143, −25.12049939706580693352950417800, −24.15277688643840534366466817387, −22.597064851967155235203517620504, −21.97297822711471949691534570438, −20.757323035700676781248438804863, −20.17829451225337709399107723389, −19.32015872986550517176570107843, −17.930851842778023517894304010579, −16.47145624245368908420169857681, −15.13145564652602594879344424775, −14.275365793319124684213026203390, −13.03041728612540454518853805872, −12.70885409887688514203595103167, −10.585140952394359219480918054833, −9.822092887607630092368112701691, −9.09260589184069462141522192547, −7.13994504195910836033294423598, −5.469012178162203515912092610376, −4.3200863026815443928955635840, −2.98888429619426650210040856374, −1.84042130931462603905989119801, 2.52753186042185668297514674442, 3.31013862521169781359768079105, 5.2630300710323405320383004613, 6.52359797869514460250958163978, 7.3120623303002138195144207355, 8.894747906095819798144701368953, 9.61016709656419464890082810976, 11.7206524304172920363395038855, 13.187682659180031387728440083, 13.56920971202157628753394798371, 14.665604458907004050747493593846, 15.65177547078027955690916167012, 16.853423472925803768811370009266, 18.16150109966303471741381829735, 18.98416253798157850737655044218, 20.40853733565497737650153301438, 21.6978117445042066982959942869, 22.16364708580796181697991199216, 23.62931464899248159846047115687, 24.825591180345275982197268058062, 25.12854498195332933935792319186, 26.44067210086640380399895210432, 26.68205357831245903645526525133, 29.10712094288516918129144826087, 29.54900717363437444733732220475

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