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Properties

Label 2-896-224.13-c0-0-0
Degree $2$
Conductor $896$
Sign $0.555 + 0.831i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
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)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (0.707 − 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (−0.292 − 0.707i)53-s − 1.00·63-s + (−0.292 + 0.707i)67-s + (−1.70 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)9-s + (−0.707 − 1.70i)11-s + (1 + i)23-s + (0.707 − 0.707i)25-s + (−0.707 + 1.70i)29-s + (0.707 − 0.292i)37-s + (0.292 + 0.707i)43-s − 1.00i·49-s + (−0.292 − 0.707i)53-s − 1.00·63-s + (−0.292 + 0.707i)67-s + (−1.70 − 0.707i)77-s + 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9684709581\)
\(L(\frac12)\) \(\approx\) \(0.9684709581\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.707 - 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (-0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T^{2} \)
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

−10.37867037697018834474307957650, −9.161401084981648883275826620811, −8.543719113023377707721609270467, −7.75667050804081421545669569929, −6.77820815604410505503678097931, −5.74965680322947126483517689282, −5.03038666312973927882021152941, −3.67321395568854535224913615701, −2.90007727478073064167155416647, −1.02917740701153812391221961772, 2.00195672710065480054624641733, 2.75027351849633254106181210761, 4.51445647198220240170724483400, 5.05266164470524463309420157174, 5.99864746304311802393021972678, 7.26823706403039517249601387275, 7.87941309732915751147963891809, 8.755953910041500744601541054870, 9.570677947728051515882716116117, 10.54301364278793173580832314525

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