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Properties

Label 2-12e3-48.29-c0-0-0
Degree $2$
Conductor $1728$
Sign $0.382 - 0.923i$
Analytic cond. $0.862384$
Root an. cond. $0.928646$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 + 0.707i)35-s + (1 − i)43-s + 1.41i·47-s + (−0.707 − 0.707i)53-s + 1.00i·55-s − 1.41i·65-s + (−1 − i)67-s + 1.41·71-s + i·73-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)5-s + i·7-s + (0.707 + 0.707i)11-s + (−1 − i)13-s + 1.41i·17-s − 31-s + (−0.707 + 0.707i)35-s + (1 − i)43-s + 1.41i·47-s + (−0.707 − 0.707i)53-s + 1.00i·55-s − 1.41i·65-s + (−1 − i)67-s + 1.41·71-s + i·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.382 - 0.923i$
Analytic conductor: \(0.862384\)
Root analytic conductor: \(0.928646\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :0),\ 0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.233905879\)
\(L(\frac12)\) \(\approx\) \(1.233905879\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + iT^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (-1 + i)T - iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + iT^{2} \)
67 \( 1 + (1 + i)T + iT^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.635267626356454775455233191156, −9.037972040293422263517966092073, −8.059338648484811005721100942922, −7.25707364344231969057443392337, −6.30731665863474286880736414604, −5.78136607915405100082449493527, −4.86704113590979641538307619655, −3.67198967629141611279574612688, −2.57643658410204539008691649513, −1.86249989552766588058839617478, 0.988546802046507834733748318804, 2.19137392873903803239592831973, 3.51119577129881542102389610905, 4.50149724821063543432988127956, 5.16747775415746864305776754321, 6.17304769351559242434115416871, 7.06455436619895528217560852122, 7.60678844659312842247690901736, 8.895893202914540090023633575803, 9.300933184979801461547204755441

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