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Properties

Label 2-1175-235.234-c0-0-5
Degree $2$
Conductor $1175$
Sign $0.447 - 0.894i$
Analytic cond. $0.586401$
Root an. cond. $0.765768$
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  + i·2-s − i·3-s + 6-s + i·7-s + i·8-s − 14-s − 16-s + i·17-s + 21-s + 24-s − i·27-s − 34-s − 2i·37-s + i·42-s − i·47-s + i·48-s + ⋯
L(s)  = 1  + i·2-s − i·3-s + 6-s + i·7-s + i·8-s − 14-s − 16-s + i·17-s + 21-s + 24-s − i·27-s − 34-s − 2i·37-s + i·42-s − i·47-s + i·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1175\)    =    \(5^{2} \cdot 47\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(0.586401\)
Root analytic conductor: \(0.765768\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1175} (1174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1175,\ (\ :0),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.199724873\)
\(L(\frac12)\) \(\approx\) \(1.199724873\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
47 \( 1 + iT \)
good2 \( 1 - iT - T^{2} \)
3 \( 1 + iT - T^{2} \)
7 \( 1 - iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
53 \( 1 - 2iT - T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + iT - T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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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

−10.02007567144953970992774214569, −8.794972811171340486941976277819, −8.383623965395212485176496519968, −7.40154684943647950945341583173, −6.95493456145418269331154294219, −5.90053447147817501586580582347, −5.63528715667107764187061336136, −4.21442734730752157624507590901, −2.60840113456155462037983586624, −1.76207488692665773861153712680, 1.21614510450567135802377798749, 2.71150939728517365815803353816, 3.65114140924175547606939483165, 4.32353904118107368100525555333, 5.20625996794060071640480626102, 6.67591095127325277900171233766, 7.22959760495157394324603483795, 8.408551159336770348340229239637, 9.597075487419223413386752739306, 9.916052497818504922253918274810

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