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Properties

Label 2-138-3.2-c2-0-1
Degree $2$
Conductor $138$
Sign $-0.842 + 0.538i$
Analytic cond. $3.76022$
Root an. cond. $1.93913$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−2.52 + 1.61i)3-s − 2.00·4-s + 7.30i·5-s + (−2.28 − 3.57i)6-s − 0.709·7-s − 2.82i·8-s + (3.77 − 8.16i)9-s − 10.3·10-s − 9.86i·11-s + (5.05 − 3.23i)12-s − 21.4·13-s − 1.00i·14-s + (−11.8 − 18.4i)15-s + 4.00·16-s + 17.8i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.842 + 0.538i)3-s − 0.500·4-s + 1.46i·5-s + (−0.380 − 0.595i)6-s − 0.101·7-s − 0.353i·8-s + (0.419 − 0.907i)9-s − 1.03·10-s − 0.897i·11-s + (0.421 − 0.269i)12-s − 1.65·13-s − 0.0717i·14-s + (−0.786 − 1.23i)15-s + 0.250·16-s + 1.05i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(138\)    =    \(2 \cdot 3 \cdot 23\)
Sign: $-0.842 + 0.538i$
Analytic conductor: \(3.76022\)
Root analytic conductor: \(1.93913\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{138} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 138,\ (\ :1),\ -0.842 + 0.538i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.155474 - 0.531845i\)
\(L(\frac12)\) \(\approx\) \(0.155474 - 0.531845i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (2.52 - 1.61i)T \)
23 \( 1 - 4.79iT \)
good5 \( 1 - 7.30iT - 25T^{2} \)
7 \( 1 + 0.709T + 49T^{2} \)
11 \( 1 + 9.86iT - 121T^{2} \)
13 \( 1 + 21.4T + 169T^{2} \)
17 \( 1 - 17.8iT - 289T^{2} \)
19 \( 1 + 8.45T + 361T^{2} \)
29 \( 1 - 4.47iT - 841T^{2} \)
31 \( 1 - 55.2T + 961T^{2} \)
37 \( 1 + 19.3T + 1.36e3T^{2} \)
41 \( 1 - 75.6iT - 1.68e3T^{2} \)
43 \( 1 + 66.6T + 1.84e3T^{2} \)
47 \( 1 - 30.7iT - 2.20e3T^{2} \)
53 \( 1 - 39.2iT - 2.80e3T^{2} \)
59 \( 1 + 1.11iT - 3.48e3T^{2} \)
61 \( 1 + 46.9T + 3.72e3T^{2} \)
67 \( 1 - 12.3T + 4.48e3T^{2} \)
71 \( 1 + 5.12iT - 5.04e3T^{2} \)
73 \( 1 - 67.8T + 5.32e3T^{2} \)
79 \( 1 - 137.T + 6.24e3T^{2} \)
83 \( 1 + 26.8iT - 6.88e3T^{2} \)
89 \( 1 - 88.0iT - 7.92e3T^{2} \)
97 \( 1 - 0.0454T + 9.40e3T^{2} \)
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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

−13.82158633057748805363900495896, −12.44251171060939489132569081046, −11.34993317812386131511492582609, −10.40883453145512771513673540668, −9.695923003583327384035007059575, −8.056031791903301064628001170541, −6.75242868979709261774178841039, −6.14446244247827477785727683199, −4.73686626179051315282524631337, −3.19918768854477156080204358096, 0.39791719529379152988371232957, 2.08584506920827691915235857456, 4.67665493787568815547486416691, 5.11857092896098796801499364807, 6.92590571690304854529135059612, 8.150358645588384017039998485831, 9.472128112924982776804485868894, 10.25718158268317846410378737655, 11.84703242213679983368932747268, 12.18751070666727925876005208186

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