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Properties

Label 2-2400-120.83-c0-0-4
Degree $2$
Conductor $2400$
Sign $0.525 + 0.850i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
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)3-s − 1.00i·9-s + (−1.41 − 1.41i)17-s − 2i·19-s + (0.707 + 0.707i)27-s − i·49-s + 2.00·51-s + (1.41 + 1.41i)57-s − 1.00·81-s + (1.41 − 1.41i)83-s + (−1.41 − 1.41i)107-s + (1.41 − 1.41i)113-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 1.00i·9-s + (−1.41 − 1.41i)17-s − 2i·19-s + (0.707 + 0.707i)27-s − i·49-s + 2.00·51-s + (1.41 + 1.41i)57-s − 1.00·81-s + (1.41 − 1.41i)83-s + (−1.41 − 1.41i)107-s + (1.41 − 1.41i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6928209460\)
\(L(\frac12)\) \(\approx\) \(0.6928209460\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
19 \( 1 + 2iT - T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT^{2} \)
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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

−9.171653848163662675633704117135, −8.526903600020311783245740846791, −7.13345846980144087644264091706, −6.81975181285477433919526399452, −5.82910223441519128053224827724, −4.81732360643643099612166989005, −4.58073494474341555171819922978, −3.32496083964286841919791289447, −2.36193084304251209585062625195, −0.51925584784192570297122705194, 1.43888144377707342146046351329, 2.26500757659459225931017918643, 3.70508193587329733889521690601, 4.52730536841506968345562110299, 5.59427009073439977881760091911, 6.19206676530015780928148700485, 6.79940384947373625385627303428, 7.84310338928159656485968237136, 8.246047028835859226879352566407, 9.202734681872988058869238199746

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