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Properties

Label 2-2e2-4.3-c42-0-6
Degree $2$
Conductor $4$
Sign $0.996 + 0.0813i$
Analytic cond. $44.6910$
Root an. cond. $6.68513$
Motivic weight $42$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.53e4 − 2.09e6i)2-s + 4.36e8i·3-s + (−4.38e12 − 3.57e11i)4-s + 5.12e14·5-s + (9.15e14 + 3.72e13i)6-s − 3.44e17i·7-s + (−1.12e18 + 9.15e18i)8-s + 1.09e20·9-s + (4.37e19 − 1.07e21i)10-s + 6.73e21i·11-s + (1.56e20 − 1.91e21i)12-s − 3.68e23·13-s + (−7.22e23 − 2.94e22i)14-s + 2.23e23i·15-s + (1.90e25 + 3.13e24i)16-s + 2.47e25·17-s + ⋯
L(s)  = 1  + (0.0407 − 0.999i)2-s + 0.0417i·3-s + (−0.996 − 0.0813i)4-s + 1.07·5-s + (0.0417 + 0.00169i)6-s − 0.617i·7-s + (−0.121 + 0.992i)8-s + 0.998·9-s + (0.0437 − 1.07i)10-s + 0.909i·11-s + (0.00339 − 0.0416i)12-s − 1.49·13-s + (−0.617 − 0.0251i)14-s + 0.0448i·15-s + (0.986 + 0.162i)16-s + 0.358·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0813i)\, \overline{\Lambda}(43-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+21) \, L(s)\cr =\mathstrut & (0.996 + 0.0813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.996 + 0.0813i$
Analytic conductor: \(44.6910\)
Root analytic conductor: \(6.68513\)
Motivic weight: \(42\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :21),\ 0.996 + 0.0813i)\)

Particular Values

\(L(\frac{43}{2})\) \(\approx\) \(2.063623847\)
\(L(\frac12)\) \(\approx\) \(2.063623847\)
\(L(22)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-8.53e4 + 2.09e6i)T \)
good3 \( 1 - 4.36e8iT - 1.09e20T^{2} \)
5 \( 1 - 5.12e14T + 2.27e29T^{2} \)
7 \( 1 + 3.44e17iT - 3.11e35T^{2} \)
11 \( 1 - 6.73e21iT - 5.47e43T^{2} \)
13 \( 1 + 3.68e23T + 6.10e46T^{2} \)
17 \( 1 - 2.47e25T + 4.77e51T^{2} \)
19 \( 1 - 1.02e27iT - 5.10e53T^{2} \)
23 \( 1 - 5.89e28iT - 1.55e57T^{2} \)
29 \( 1 - 5.42e29T + 2.63e61T^{2} \)
31 \( 1 - 4.43e30iT - 4.33e62T^{2} \)
37 \( 1 - 2.65e32T + 7.31e65T^{2} \)
41 \( 1 - 2.58e33T + 5.45e67T^{2} \)
43 \( 1 - 2.21e34iT - 4.03e68T^{2} \)
47 \( 1 + 1.77e35iT - 1.69e70T^{2} \)
53 \( 1 - 3.00e36T + 2.62e72T^{2} \)
59 \( 1 + 1.17e37iT - 2.37e74T^{2} \)
61 \( 1 - 1.65e37T + 9.63e74T^{2} \)
67 \( 1 - 3.76e38iT - 4.95e76T^{2} \)
71 \( 1 + 5.25e38iT - 5.66e77T^{2} \)
73 \( 1 + 9.87e38T + 1.81e78T^{2} \)
79 \( 1 - 3.24e39iT - 5.01e79T^{2} \)
83 \( 1 - 2.03e40iT - 3.99e80T^{2} \)
89 \( 1 + 3.29e40T + 7.48e81T^{2} \)
97 \( 1 + 8.36e41T + 2.78e83T^{2} \)
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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

−14.63147331137899420875726564168, −13.33078265888734869552431892737, −12.17292549289744892462842911702, −10.06221085591678047677050470274, −9.788303006573567091545845385459, −7.43167685282260773188483075307, −5.32607785138536357126629173979, −3.98684599618177333380526336609, −2.19127021687743111144758044673, −1.26730866151655992460591007752, 0.60574323235229598544667640844, 2.48949419855483669583901822949, 4.65636105473958820495289534824, 5.86749827615759226703682415409, 7.16185607558894956304583498256, 8.911651849458960707313615045764, 10.05162871905454259614034139531, 12.58159928278551847056169331379, 13.81334446986510933971145987355, 15.14612933853412758229116407736

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