Properties

Label 2-2240-140.39-c0-0-1
Degree $2$
Conductor $2240$
Sign $0.553 - 0.832i$
Analytic cond. $1.11790$
Root an. cond. $1.05731$
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  + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)9-s + 1.73·15-s + (1.5 − 0.866i)21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s + 29-s + 0.999i·35-s + 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 + 1.5i)3-s + (−0.5 − 0.866i)5-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)9-s + 1.73·15-s + (1.5 − 0.866i)21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 1.73·27-s + 29-s + 0.999i·35-s + 41-s + 1.73·43-s + (−1 + 1.73i)45-s + (0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2240\)    =    \(2^{6} \cdot 5 \cdot 7\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(1.11790\)
Root analytic conductor: \(1.05731\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2240} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2240,\ (\ :0),\ 0.553 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6458266478\)
\(L(\frac12)\) \(\approx\) \(0.6458266478\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T + T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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

−9.393030097162222583668658239065, −8.976129693503583186396641634054, −7.84453930217523303237642455877, −6.93141127189119104190415844273, −5.90702926977620149401140359372, −5.31402991040925593567164498904, −4.44835271220995208030160186489, −3.90820905313702025192651192642, −3.06825678540001141763926428415, −0.872306420878543368551202482584, 0.72813443902460122158007661839, 2.34637216350343368186167014413, 2.92055652798516423600258488818, 4.29030958929265262708550854343, 5.50270074426703396038544367209, 6.24675636203387813849334724495, 6.73441257555937397827771000009, 7.28901103937944453628923985106, 8.121444407704655220593378752923, 8.900156357031772046853475391740

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