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Properties

Label 2-189-189.5-c1-0-17
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: $\chi_{189} (5, \cdot )$
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

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

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