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Properties

Label 2-1170-65.47-c1-0-22
Degree $2$
Conductor $1170$
Sign $-0.256 + 0.966i$
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 − 2·7-s + i·8-s + (−1 − 2i)10-s + (−1 + i)11-s + (2 + 3i)13-s + 2i·14-s + 16-s + (5 − 5i)17-s + (3 − 3i)19-s + (−2 + i)20-s + (1 + i)22-s + (−5 − 5i)23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.894 − 0.447i)5-s − 0.755·7-s + 0.353i·8-s + (−0.316 − 0.632i)10-s + (−0.301 + 0.301i)11-s + (0.554 + 0.832i)13-s + 0.534i·14-s + 0.250·16-s + (1.21 − 1.21i)17-s + (0.688 − 0.688i)19-s + (−0.447 + 0.223i)20-s + (0.213 + 0.213i)22-s + (−1.04 − 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1170\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.256 + 0.966i$
Analytic conductor: \(9.34249\)
Root analytic conductor: \(3.05654\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1170} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1170,\ (\ :1/2),\ -0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.617963774\)
\(L(\frac12)\) \(\approx\) \(1.617963774\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 3i)T \)
good7 \( 1 + 2T + 7T^{2} \)
11 \( 1 + (1 - i)T - 11iT^{2} \)
17 \( 1 + (-5 + 5i)T - 17iT^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + (-1 - i)T + 31iT^{2} \)
37 \( 1 - 8T + 37T^{2} \)
41 \( 1 + (1 + i)T + 41iT^{2} \)
43 \( 1 + (5 + 5i)T + 43iT^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + (-3 - 3i)T + 59iT^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 + (1 + i)T + 71iT^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + (7 + 7i)T + 89iT^{2} \)
97 \( 1 + 2iT - 97T^{2} \)
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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

−9.811838175048786780602413270653, −9.030261698202166964057755170758, −8.109849688819639953413362955277, −6.95929710638061319891645237533, −6.06654771639749607906063752560, −5.19128808909804337791728829105, −4.28078355813018062615844161158, −3.06211118418067841897589514107, −2.15952476808697406578861119602, −0.77372540478082742122263394944, 1.38243838830706050273620078512, 3.05946393735123265380633758987, 3.73232045753334621310373782290, 5.37574314746323830061995415432, 5.87943034653424371510792575730, 6.44425569459056479094168617265, 7.64749827351978548361245265838, 8.154083070485662014745403370583, 9.286599700174008690944272622730, 10.05288630063423708510297711997

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