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Properties

Label 2-3840-480.269-c0-0-0
Degree $2$
Conductor $3840$
Sign $0.980 + 0.195i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)5-s + (−0.707 − 0.707i)9-s + i·15-s + 1.84i·17-s + (0.707 + 0.292i)19-s + (1.30 + 1.30i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 1.41·31-s + (0.923 + 0.382i)45-s − 0.765i·47-s − i·49-s + (1.70 + 0.707i)51-s + (−0.541 − 1.30i)53-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)3-s + (−0.923 + 0.382i)5-s + (−0.707 − 0.707i)9-s + i·15-s + 1.84i·17-s + (0.707 + 0.292i)19-s + (1.30 + 1.30i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)27-s + 1.41·31-s + (0.923 + 0.382i)45-s − 0.765i·47-s − i·49-s + (1.70 + 0.707i)51-s + (−0.541 − 1.30i)53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.980 + 0.195i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ 0.980 + 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221006759\)
\(L(\frac12)\) \(\approx\) \(1.221006759\)
\(L(1)\) 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 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.923 - 0.382i)T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 - 1.84iT - T^{2} \)
19 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
29 \( 1 + (0.707 + 0.707i)T^{2} \)
31 \( 1 - 1.41T + T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.707 - 0.707i)T^{2} \)
47 \( 1 + 0.765iT - T^{2} \)
53 \( 1 + (0.541 + 1.30i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 + 0.707i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−8.263821703245773154602297545896, −8.066084784631364851740995673502, −7.17749339467015105773997161213, −6.65105813940284318908669959883, −5.85603023785228742956625700833, −4.88659226538389374117700570336, −3.63069446431619706087755631892, −3.35613810969334162510469332908, −2.14883118490354350542857998179, −1.07681072416738276678013071308, 0.841470044982524098886367939067, 2.78867857836837900263847135676, 3.04580299743188257617335767107, 4.33820095581080924616287685922, 4.69891718221060353794100120682, 5.37672155741018468800005431033, 6.57990825502235017091669183512, 7.43722041204596652906833629190, 7.971702500072484391009444011389, 8.937362713752551167820455632739

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