Properties

Label 2-114-57.50-c1-0-4
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: $\chi_{114} (107, \cdot )$
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

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

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