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Properties

Label 2-114-19.7-c1-0-1
Degree $2$
Conductor $114$
Sign $0.321 - 0.946i$
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 + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + 7-s − 0.999·8-s + (−0.499 + 0.866i)9-s − 2·11-s − 0.999·12-s + (1.5 − 2.59i)13-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s + (−2 − 3.46i)17-s − 0.999·18-s + (4 − 1.73i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s + 0.377·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s − 0.603·11-s − 0.288·12-s + (0.416 − 0.720i)13-s + (0.133 + 0.231i)14-s + (−0.125 − 0.216i)16-s + (−0.485 − 0.840i)17-s − 0.235·18-s + (0.917 − 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05235 + 0.753654i\)
\(L(\frac12)\) \(\approx\) \(1.05235 + 0.753654i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-4 + 1.73i)T \)
good5 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 3T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + (2 + 3.46i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.5 - 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (7 + 12.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 8T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−13.88935122386104145733075453711, −13.09184942614252579263172866433, −11.75746320541781387013962392169, −10.63941498532250438456701673974, −9.403497119786081352456293968974, −8.280834324274769893054065768919, −7.28796894625811906590406858420, −5.68181664238609595037831782148, −4.63444084371942773196590408353, −3.03965187205450842041976789976, 1.88720241946061314769023443231, 3.61870851811828748658304663038, 5.19544346410358851280221525081, 6.63182940809195906115128037212, 8.042395623129522344473548287943, 9.113955766948714279850299130543, 10.44669732874482476927455045214, 11.41664737948096430275132660373, 12.42716835803818488674609293872, 13.33372181003026943671910859257

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