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Properties

Label 2-3920-140.87-c0-0-1
Degree $2$
Conductor $3920$
Sign $0.958 + 0.286i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  + (0.608 − 0.793i)5-s + (0.866 − 0.5i)9-s + (1.30 + 1.30i)13-s + (0.198 + 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s + 1.84i·41-s + (0.130 − 0.991i)45-s + (−0.366 − 1.36i)53-s + (−0.662 + 0.382i)61-s + (1.83 − 0.241i)65-s + (−0.739 + 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)5-s + (0.866 − 0.5i)9-s + (1.30 + 1.30i)13-s + (0.198 + 0.739i)17-s + (−0.258 − 0.965i)25-s − 1.41i·29-s + (−0.517 + 1.93i)37-s + 1.84i·41-s + (0.130 − 0.991i)45-s + (−0.366 − 1.36i)53-s + (−0.662 + 0.382i)61-s + (1.83 − 0.241i)65-s + (−0.739 + 0.198i)73-s + (0.499 − 0.866i)81-s + (0.707 + 0.292i)85-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.958 + 0.286i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (3167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :0),\ 0.958 + 0.286i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.674440431\)
\(L(\frac12)\) \(\approx\) \(1.674440431\)
\(L(1)\) 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 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.662 - 0.382i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
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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

−8.470930533108821281231982259987, −8.214620851617796815037196807236, −6.95050829907999220654605834447, −6.33456887657352719276813142796, −5.84049881265685798933031272334, −4.60038746118949130947837535324, −4.25437859252723123202701610097, −3.24565989321870136426878551485, −1.80036553905066311931658170668, −1.28301217238724581554223030146, 1.23375566306066413544447781518, 2.28029721270195856769038844325, 3.24053214140901345757023793256, 3.91415239483954902311765517159, 5.17210373226385469999388904761, 5.64305920756410596978973490187, 6.50094030563071988560733070827, 7.30864893673542978583929611170, 7.71022234698632372835050622984, 8.803267991805535863481753377741

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