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Properties

Label 2-114-57.50-c1-0-3
Degree $2$
Conductor $114$
Sign $0.740 + 0.671i$
Analytic cond. $0.910294$
Root an. cond. $0.954093$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (3 − 1.73i)5-s + (1.5 + 0.866i)6-s + 7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (−3 − 1.73i)10-s − 3.46i·11-s − 1.73i·12-s + (4.5 + 2.59i)13-s + (−0.5 − 0.866i)14-s + (−3 + 5.19i)15-s + (−0.5 − 0.866i)16-s + (−3 + 1.73i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (1.34 − 0.774i)5-s + (0.612 + 0.353i)6-s + 0.377·7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (−0.948 − 0.547i)10-s − 1.04i·11-s − 0.499i·12-s + (1.24 + 0.720i)13-s + (−0.133 − 0.231i)14-s + (−0.774 + 1.34i)15-s + (−0.125 − 0.216i)16-s + (−0.727 + 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.790445 - 0.304972i\)
\(L(\frac12)\) \(\approx\) \(0.790445 - 0.304972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
19 \( 1 + (4 - 1.73i)T \)
good5 \( 1 + (-3 + 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + (-4.5 - 2.59i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3 - 1.73i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 5.19iT - 37T^{2} \)
41 \( 1 + (6 + 10.3i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 - 3.46i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6 - 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.5 - 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.3iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6 + 3.46i)T + (48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.27434760119080765523981556636, −12.37072696131862652715569093396, −11.04861214084153788955896342041, −10.59060387862236619016136009864, −9.186213420805656112654786863573, −8.684846358385062173216292270885, −6.39482164787404369910238904729, −5.45950109600893872042994927731, −4.03354411211094901129353740907, −1.54712672288837368532835762924, 1.95603494038211965070339268295, 4.91125671332175555338440680923, 6.15889112416089991581995474411, 6.73473680322314757227157679596, 8.085225391399459615181427699381, 9.632275193483914585235363904954, 10.52385192800133151758798951739, 11.36889655736457676151856995986, 13.06125188585223144672285340562, 13.53655287403997642484764081269

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