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Properties

Label 2-114-57.8-c1-0-5
Degree $2$
Conductor $114$
Sign $0.689 + 0.724i$
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 + (1.72 + 0.158i)3-s + (−0.499 − 0.866i)4-s + (−1.22 − 0.707i)5-s + (1 − 1.41i)6-s − 0.449·7-s − 0.999·8-s + (2.94 + 0.548i)9-s + (−1.22 + 0.707i)10-s + 3.14i·11-s + (−0.724 − 1.57i)12-s + (−3 + 1.73i)13-s + (−0.224 + 0.389i)14-s + (−1.99 − 1.41i)15-s + (−0.5 + 0.866i)16-s + (−0.550 − 0.317i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.995 + 0.0917i)3-s + (−0.249 − 0.433i)4-s + (−0.547 − 0.316i)5-s + (0.408 − 0.577i)6-s − 0.169·7-s − 0.353·8-s + (0.983 + 0.182i)9-s + (−0.387 + 0.223i)10-s + 0.948i·11-s + (−0.209 − 0.454i)12-s + (−0.832 + 0.480i)13-s + (−0.0600 + 0.104i)14-s + (−0.516 − 0.365i)15-s + (−0.125 + 0.216i)16-s + (−0.133 − 0.0770i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(114\)    =    \(2 \cdot 3 \cdot 19\)
Sign: $0.689 + 0.724i$
Analytic conductor: \(0.910294\)
Root analytic conductor: \(0.954093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{114} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 114,\ (\ :1/2),\ 0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31380 - 0.562999i\)
\(L(\frac12)\) \(\approx\) \(1.31380 - 0.562999i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.158i)T \)
19 \( 1 + (-3.17 + 2.98i)T \)
good5 \( 1 + (1.22 + 0.707i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 0.449T + 7T^{2} \)
11 \( 1 - 3.14iT - 11T^{2} \)
13 \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.550 + 0.317i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.22 - 2.12i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.24iT - 31T^{2} \)
37 \( 1 - 7.70iT - 37T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.44 + 7.70i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-11.5 + 6.68i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.44 + 9.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.72 + 9.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.775 - 1.34i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.17 + 1.25i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (4.39 - 7.61i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.34 + 4.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + (-3.55 - 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.84 - 1.64i)T + (48.5 + 84.0i)T^{2} \)
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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

−13.47785012357537188023401145169, −12.41332424390765453096914202034, −11.67033132396131967854051039721, −10.04563987942332538864544266026, −9.443051614271687020926982601934, −8.119244971647420069877070792288, −7.00212381133974054533424786471, −4.88440965723369080757589943641, −3.82709755954716876771361966832, −2.23429147108818505649651132313, 2.95364971832215458448376476736, 4.14371224039697263635162358299, 5.91312693089997754829289866918, 7.37587048204865065347075386722, 8.038604944968059247643151268766, 9.206538024876112526840697440671, 10.44680927117886313211454587860, 11.97382812889269244257809344125, 12.89063116461587249338652393960, 14.07636035207245084414248579914

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