Properties

Label 2-550-5.4-c5-0-68
Degree $2$
Conductor $550$
Sign $-0.894 - 0.447i$
Analytic cond. $88.2111$
Root an. cond. $9.39207$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s i·3-s − 16·4-s − 4·6-s − 166i·7-s + 64i·8-s + 242·9-s − 121·11-s + 16i·12-s − 692i·13-s − 664·14-s + 256·16-s − 738i·17-s − 968i·18-s − 1.42e3·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.0641i·3-s − 0.5·4-s − 0.0453·6-s − 1.28i·7-s + 0.353i·8-s + 0.995·9-s − 0.301·11-s + 0.0320i·12-s − 1.13i·13-s − 0.905·14-s + 0.250·16-s − 0.619i·17-s − 0.704i·18-s − 0.904·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(550\)    =    \(2 \cdot 5^{2} \cdot 11\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(88.2111\)
Root analytic conductor: \(9.39207\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{550} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 550,\ (\ :5/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.423973614\)
\(L(\frac12)\) \(\approx\) \(1.423973614\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
5 \( 1 \)
11 \( 1 + 121T \)
good3 \( 1 + iT - 243T^{2} \)
7 \( 1 + 166iT - 1.68e4T^{2} \)
13 \( 1 + 692iT - 3.71e5T^{2} \)
17 \( 1 + 738iT - 1.41e6T^{2} \)
19 \( 1 + 1.42e3T + 2.47e6T^{2} \)
23 \( 1 - 1.77e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.06e3T + 2.05e7T^{2} \)
31 \( 1 - 6.24e3T + 2.86e7T^{2} \)
37 \( 1 + 1.47e4iT - 6.93e7T^{2} \)
41 \( 1 - 5.30e3T + 1.15e8T^{2} \)
43 \( 1 + 1.77e4iT - 1.47e8T^{2} \)
47 \( 1 + 1.71e4iT - 2.29e8T^{2} \)
53 \( 1 - 3.07e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.49e4T + 7.14e8T^{2} \)
61 \( 1 + 4.59e4T + 8.44e8T^{2} \)
67 \( 1 - 2.53e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.33e4T + 1.80e9T^{2} \)
73 \( 1 - 5.32e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.72e4T + 3.07e9T^{2} \)
83 \( 1 + 5.50e4iT - 3.93e9T^{2} \)
89 \( 1 + 1.25e5T + 5.58e9T^{2} \)
97 \( 1 + 8.88e4iT - 8.58e9T^{2} \)
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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

−9.902828077097590753236647029353, −8.690529889178830796639051056589, −7.62282010702504340294586630868, −7.03989666314006893544576883718, −5.62666795124047377613043516883, −4.47206890521511149385129833377, −3.76786314485316018272266988607, −2.52412045818749208559136438371, −1.17284060619214362295062690224, −0.34790198358547525806986581996, 1.47190582118652475489307462558, 2.67726972979210351509070079934, 4.24859334485509548506669224700, 4.89541214036486517515099927716, 6.24525504963952091761986971035, 6.61906687308860739010336821652, 7.973086720598388884943927681493, 8.616530177468983034762567960627, 9.503289496218227891152006339862, 10.26772840484136251331668946053

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