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Properties

Label 2-832-52.47-c1-0-13
Degree $2$
Conductor $832$
Sign $0.772 - 0.635i$
Analytic cond. $6.64355$
Root an. cond. $2.57750$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3 + 3i)5-s + 3·9-s + (2 − 3i)13-s − 2i·17-s + 13i·25-s − 4·29-s + (−5 + 5i)37-s + (−1 − i)41-s + (9 + 9i)45-s − 7i·49-s + 14·53-s − 10·61-s + (15 − 3i)65-s + (−11 + 11i)73-s + 9·81-s + ⋯
L(s)  = 1  + (1.34 + 1.34i)5-s + 9-s + (0.554 − 0.832i)13-s − 0.485i·17-s + 2.60i·25-s − 0.742·29-s + (−0.821 + 0.821i)37-s + (−0.156 − 0.156i)41-s + (1.34 + 1.34i)45-s − i·49-s + 1.92·53-s − 1.28·61-s + (1.86 − 0.372i)65-s + (−1.28 + 1.28i)73-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.772 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(832\)    =    \(2^{6} \cdot 13\)
Sign: $0.772 - 0.635i$
Analytic conductor: \(6.64355\)
Root analytic conductor: \(2.57750\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{832} (255, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 832,\ (\ :1/2),\ 0.772 - 0.635i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00095 + 0.717726i\)
\(L(\frac12)\) \(\approx\) \(2.00095 + 0.717726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (-2 + 3i)T \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + (-3 - 3i)T + 5iT^{2} \)
7 \( 1 + 7iT^{2} \)
11 \( 1 + 11iT^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (5 - 5i)T - 37iT^{2} \)
41 \( 1 + (1 + i)T + 41iT^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 14T + 53T^{2} \)
59 \( 1 + 59iT^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71iT^{2} \)
73 \( 1 + (11 - 11i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-3 + 3i)T - 89iT^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−10.25437543733590830410422149246, −9.764383010643347503038070439514, −8.761724540546358396259063182468, −7.45498602611460800887500202683, −6.86923149048460101023629218517, −6.00360825524516310341230000612, −5.19860103927476500618244190682, −3.70749005878902586892220257092, −2.71410711079496679402404178008, −1.58650910479118563036547618736, 1.27847269822031012200981449224, 2.06267887484507689610796891161, 3.94556229738373482275029963228, 4.77857940341211382052236648333, 5.71499546795686880561942593312, 6.47383511891298247904067413358, 7.58515108176523553304441013961, 8.796903204482485740554965196373, 9.160351301620663488834892388959, 10.01380297483268663077915706474

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