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Properties

Label 2-189-189.38-c1-0-15
Degree $2$
Conductor $189$
Sign $0.937 - 0.349i$
Analytic cond. $1.50917$
Root an. cond. $1.22848$
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  + (1.72 + 0.304i)2-s + (1.65 + 0.499i)3-s + (1.01 + 0.370i)4-s + (−0.0612 − 0.347i)5-s + (2.71 + 1.36i)6-s + (−2.64 − 0.150i)7-s + (−1.39 − 0.805i)8-s + (2.50 + 1.65i)9-s − 0.618i·10-s + (0.991 + 0.174i)11-s + (1.50 + 1.12i)12-s + (−0.216 − 0.258i)13-s + (−4.52 − 1.06i)14-s + (0.0717 − 0.606i)15-s + (−3.82 − 3.20i)16-s − 5.22·17-s + ⋯
L(s)  = 1  + (1.22 + 0.215i)2-s + (0.957 + 0.288i)3-s + (0.508 + 0.185i)4-s + (−0.0273 − 0.155i)5-s + (1.10 + 0.558i)6-s + (−0.998 − 0.0567i)7-s + (−0.493 − 0.284i)8-s + (0.833 + 0.551i)9-s − 0.195i·10-s + (0.299 + 0.0527i)11-s + (0.433 + 0.323i)12-s + (−0.0600 − 0.0715i)13-s + (−1.20 − 0.284i)14-s + (0.0185 − 0.156i)15-s + (−0.956 − 0.802i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(189\)    =    \(3^{3} \cdot 7\)
Sign: $0.937 - 0.349i$
Analytic conductor: \(1.50917\)
Root analytic conductor: \(1.22848\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{189} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 189,\ (\ :1/2),\ 0.937 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31722 + 0.417735i\)
\(L(\frac12)\) \(\approx\) \(2.31722 + 0.417735i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 - 0.499i)T \)
7 \( 1 + (2.64 + 0.150i)T \)
good2 \( 1 + (-1.72 - 0.304i)T + (1.87 + 0.684i)T^{2} \)
5 \( 1 + (0.0612 + 0.347i)T + (-4.69 + 1.71i)T^{2} \)
11 \( 1 + (-0.991 - 0.174i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (0.216 + 0.258i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + 5.22T + 17T^{2} \)
19 \( 1 - 0.554iT - 19T^{2} \)
23 \( 1 + (-1.52 - 1.81i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.750 + 0.893i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.597 - 1.64i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.07 + 5.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (8.82 - 7.40i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-5.78 + 2.10i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-5.19 + 1.88i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + (-3.63 - 2.10i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.1 - 8.51i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (4.82 + 13.2i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (-1.45 - 8.25i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (9.96 - 5.75i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.89 + 3.97i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.05 + 11.6i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-2.81 - 2.36i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 - 3.99T + 89T^{2} \)
97 \( 1 + (-5.45 - 14.9i)T + (-74.3 + 62.3i)T^{2} \)
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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

−13.01119417592905550747859661119, −12.12624962180681433764222460125, −10.63584651077380853428224541250, −9.439192560726753426127107691996, −8.808178864728483414439769748813, −7.21459113210756775535254924918, −6.25861526776930182131635097392, −4.79857182200407486797239782968, −3.83437413887676720562410677014, −2.74639514985581887733756317935, 2.52111318107597492712608198647, 3.50005731502664630786715743580, 4.60433782306069083517245711777, 6.24131930865771214350709889221, 7.04736656775189715803775796063, 8.672592455615827120682751927198, 9.321145073792492569721723835032, 10.70977347936077001989816164931, 12.00549426947225368758313364003, 12.80882675370763814464918556460

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