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Properties

Label 2-2904-2904.797-c0-0-1
Degree $2$
Conductor $2904$
Sign $-0.304 - 0.952i$
Analytic cond. $1.44928$
Root an. cond. $1.20386$
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  + (0.897 − 0.441i)2-s + (−0.809 − 0.587i)3-s + (0.610 − 0.791i)4-s + (−1.38 − 1.27i)5-s + (−0.985 − 0.170i)6-s + (−0.617 + 0.0352i)7-s + (0.198 − 0.980i)8-s + (0.309 + 0.951i)9-s + (−1.80 − 0.530i)10-s + (−0.959 − 0.281i)11-s + (−0.959 + 0.281i)12-s + (−0.538 + 0.303i)14-s + (0.374 + 1.84i)15-s + (−0.254 − 0.967i)16-s + (0.696 + 0.717i)18-s + ⋯
L(s)  = 1  + (0.897 − 0.441i)2-s + (−0.809 − 0.587i)3-s + (0.610 − 0.791i)4-s + (−1.38 − 1.27i)5-s + (−0.985 − 0.170i)6-s + (−0.617 + 0.0352i)7-s + (0.198 − 0.980i)8-s + (0.309 + 0.951i)9-s + (−1.80 − 0.530i)10-s + (−0.959 − 0.281i)11-s + (−0.959 + 0.281i)12-s + (−0.538 + 0.303i)14-s + (0.374 + 1.84i)15-s + (−0.254 − 0.967i)16-s + (0.696 + 0.717i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2904\)    =    \(2^{3} \cdot 3 \cdot 11^{2}\)
Sign: $-0.304 - 0.952i$
Analytic conductor: \(1.44928\)
Root analytic conductor: \(1.20386\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2904} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2904,\ (\ :0),\ -0.304 - 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4469705606\)
\(L(\frac12)\) \(\approx\) \(0.4469705606\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.897 + 0.441i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.959 + 0.281i)T \)
good5 \( 1 + (1.38 + 1.27i)T + (0.0855 + 0.996i)T^{2} \)
7 \( 1 + (0.617 - 0.0352i)T + (0.993 - 0.113i)T^{2} \)
13 \( 1 + (0.362 - 0.931i)T^{2} \)
17 \( 1 + (-0.696 + 0.717i)T^{2} \)
19 \( 1 + (-0.897 + 0.441i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 + (0.149 - 1.73i)T + (-0.985 - 0.170i)T^{2} \)
31 \( 1 + (0.0567 + 1.98i)T + (-0.998 + 0.0570i)T^{2} \)
37 \( 1 + (-0.774 + 0.633i)T^{2} \)
41 \( 1 + (0.564 + 0.825i)T^{2} \)
43 \( 1 + (0.654 - 0.755i)T^{2} \)
47 \( 1 + (0.0285 + 0.999i)T^{2} \)
53 \( 1 + (-0.129 + 0.491i)T + (-0.870 - 0.491i)T^{2} \)
59 \( 1 + (-0.859 - 1.62i)T + (-0.564 + 0.825i)T^{2} \)
61 \( 1 + (-0.610 - 0.791i)T^{2} \)
67 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.466 - 0.884i)T^{2} \)
73 \( 1 + (0.0620 - 0.159i)T + (-0.736 - 0.676i)T^{2} \)
79 \( 1 + (1.98 + 0.227i)T + (0.974 + 0.226i)T^{2} \)
83 \( 1 + (-1.07 + 0.882i)T + (0.198 - 0.980i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.687 + 0.631i)T + (0.0855 - 0.996i)T^{2} \)
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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

−8.220530982141877276780992419654, −7.52934006600469276674965587957, −6.88885898019932899058940739925, −5.79639095597328923430625368417, −5.29361337312667958161383819711, −4.54152660806431236663336537756, −3.81132970811606431792871663867, −2.75995409308096658939278802176, −1.38695612731640930377405326324, −0.23198905238877317662262641199, 2.61650430383929814969467203886, 3.35360393623308324488914296941, 3.99143301712886133078577219861, 4.76239151799207002182236506985, 5.62944638180697857125587264690, 6.58734077117395520822408488414, 6.88769033129633083200667300552, 7.73300980105071455750163837573, 8.362613548208401844843821609569, 9.701148426750989868540457353868

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