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Properties

Label 2-2e2-4.3-c38-0-12
Degree $2$
Conductor $4$
Sign $0.781 + 0.623i$
Analytic cond. $36.5853$
Root an. cond. $6.04858$
Motivic weight $38$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.73e5 + 4.94e5i)2-s − 8.74e8i·3-s + (−2.14e11 − 1.71e11i)4-s + 2.48e13·5-s + (4.32e14 + 1.51e14i)6-s − 8.29e15i·7-s + (1.22e17 − 7.66e16i)8-s + 5.86e17·9-s + (−4.30e18 + 1.23e19i)10-s + 3.43e19i·11-s + (−1.49e20 + 1.87e20i)12-s + 9.38e20·13-s + (4.10e21 + 1.43e21i)14-s − 2.17e22i·15-s + (1.68e22 + 7.36e22i)16-s − 2.90e23·17-s + ⋯
L(s)  = 1  + (−0.330 + 0.943i)2-s − 0.752i·3-s + (−0.781 − 0.623i)4-s + 1.30·5-s + (0.709 + 0.248i)6-s − 0.727i·7-s + (0.846 − 0.531i)8-s + 0.434·9-s + (−0.430 + 1.23i)10-s + 0.561i·11-s + (−0.468 + 0.587i)12-s + 0.642·13-s + (0.686 + 0.240i)14-s − 0.980i·15-s + (0.222 + 0.974i)16-s − 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(36.5853\)
Root analytic conductor: \(6.04858\)
Motivic weight: \(38\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :19),\ 0.781 + 0.623i)\)

Particular Values

\(L(\frac{39}{2})\) \(\approx\) \(2.044263414\)
\(L(\frac12)\) \(\approx\) \(2.044263414\)
\(L(20)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.73e5 - 4.94e5i)T \)
good3 \( 1 + 8.74e8iT - 1.35e18T^{2} \)
5 \( 1 - 2.48e13T + 3.63e26T^{2} \)
7 \( 1 + 8.29e15iT - 1.29e32T^{2} \)
11 \( 1 - 3.43e19iT - 3.74e39T^{2} \)
13 \( 1 - 9.38e20T + 2.13e42T^{2} \)
17 \( 1 + 2.90e23T + 5.71e46T^{2} \)
19 \( 1 + 3.13e24iT - 3.91e48T^{2} \)
23 \( 1 - 4.99e25iT - 5.56e51T^{2} \)
29 \( 1 - 8.62e27T + 3.72e55T^{2} \)
31 \( 1 - 1.05e28iT - 4.69e56T^{2} \)
37 \( 1 + 4.26e29T + 3.90e59T^{2} \)
41 \( 1 - 8.28e30T + 1.93e61T^{2} \)
43 \( 1 + 3.48e29iT - 1.17e62T^{2} \)
47 \( 1 + 1.09e32iT - 3.46e63T^{2} \)
53 \( 1 + 3.01e32T + 3.33e65T^{2} \)
59 \( 1 + 1.34e33iT - 1.96e67T^{2} \)
61 \( 1 + 2.87e33T + 6.95e67T^{2} \)
67 \( 1 + 9.77e33iT - 2.45e69T^{2} \)
71 \( 1 + 2.88e35iT - 2.22e70T^{2} \)
73 \( 1 + 3.46e35T + 6.40e70T^{2} \)
79 \( 1 + 4.41e35iT - 1.28e72T^{2} \)
83 \( 1 + 3.81e36iT - 8.41e72T^{2} \)
89 \( 1 + 3.10e36T + 1.19e74T^{2} \)
97 \( 1 - 4.23e37T + 3.14e75T^{2} \)
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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

−15.68604241199869674383946570447, −13.78551837047646006124026708716, −13.20123930143656307060217111998, −10.40914138105307628894542679141, −9.048474093819234283160839219546, −7.20622254847774554674476108522, −6.34471349578663597110331122938, −4.64953595046113778372756172302, −1.90341542259816805398329168195, −0.73967707676142316009288837285, 1.35658477420059944512227007034, 2.61368438815853361796231299947, 4.27179211635594211495284348581, 5.89372737756015670003405202900, 8.661321090706403494128493665199, 9.754892206213271337552148593922, 10.83009827454894126946204277712, 12.68630185738472199633888396426, 14.06166691340771099299209310142, 16.11813712634270594170429411231

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