Properties

Label 2-1170-5.4-c1-0-10
Degree $2$
Conductor $1170$
Sign $0.894 + 0.447i$
Analytic cond. $9.34249$
Root an. cond. $3.05654$
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  i·2-s − 4-s + (−2 − i)5-s + 2i·7-s + i·8-s + (−1 + 2i)10-s − 2·11-s + i·13-s + 2·14-s + 16-s − 2i·17-s + 4·19-s + (2 + i)20-s + 2i·22-s + (3 + 4i)25-s + 26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.894 − 0.447i)5-s + 0.755i·7-s + 0.353i·8-s + (−0.316 + 0.632i)10-s − 0.603·11-s + 0.277i·13-s + 0.534·14-s + 0.250·16-s − 0.485i·17-s + 0.917·19-s + (0.447 + 0.223i)20-s + 0.426i·22-s + (0.600 + 0.800i)25-s + 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.178112087\)
\(L(\frac12)\) \(\approx\) \(1.178112087\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
5 \( 1 + (2 + i)T \)
13 \( 1 - iT \)
good7 \( 1 - 2iT - 7T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 2iT - 53T^{2} \)
59 \( 1 - 10T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 16iT - 67T^{2} \)
71 \( 1 - 4T + 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−9.654174481263098400305920372138, −8.997031072783284288748607379997, −8.171815759576235563455702877760, −7.53615272390464044425244229702, −6.28144202217683831298153005352, −5.12909958999602973102193775545, −4.56485444308437704603388022604, −3.34827504757729037250009898440, −2.50507709629848732940770638817, −0.930918142397487302835950278504, 0.72864149967538039579016075059, 2.81451474276305788264398277562, 3.84389128685193229933124713034, 4.63486277788420801135174057666, 5.69199562201546320598422739492, 6.70138623465424849284267852003, 7.42680077058700292471204770823, 7.977949416082132357355638043698, 8.744488276021039587369213120821, 9.979970515295386925813278834114

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