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Properties

Label 2-1218-203.41-c1-0-35
Degree $2$
Conductor $1218$
Sign $0.987 + 0.159i$
Analytic cond. $9.72577$
Root an. cond. $3.11861$
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  + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + 3.88·5-s + 1.00·6-s + (−0.687 − 2.55i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (2.74 + 2.74i)10-s + (−2.40 − 2.40i)11-s + (0.707 + 0.707i)12-s + 1.76·13-s + (1.32 − 2.29i)14-s + (2.74 − 2.74i)15-s − 1.00·16-s + (−2.16 + 2.16i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + 1.73·5-s + 0.408·6-s + (−0.259 − 0.965i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.867 + 0.867i)10-s + (−0.724 − 0.724i)11-s + (0.204 + 0.204i)12-s + 0.490·13-s + (0.352 − 0.612i)14-s + (0.708 − 0.708i)15-s − 0.250·16-s + (−0.525 + 0.525i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1218 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1218\)    =    \(2 \cdot 3 \cdot 7 \cdot 29\)
Sign: $0.987 + 0.159i$
Analytic conductor: \(9.72577\)
Root analytic conductor: \(3.11861\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1218} (853, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1218,\ (\ :1/2),\ 0.987 + 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.169054931\)
\(L(\frac12)\) \(\approx\) \(3.169054931\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.687 + 2.55i)T \)
29 \( 1 + (-3.09 + 4.40i)T \)
good5 \( 1 - 3.88T + 5T^{2} \)
11 \( 1 + (2.40 + 2.40i)T + 11iT^{2} \)
13 \( 1 - 1.76T + 13T^{2} \)
17 \( 1 + (2.16 - 2.16i)T - 17iT^{2} \)
19 \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \)
23 \( 1 - 7.16T + 23T^{2} \)
31 \( 1 + (0.785 - 0.785i)T - 31iT^{2} \)
37 \( 1 + (6.21 - 6.21i)T - 37iT^{2} \)
41 \( 1 + (-0.510 - 0.510i)T + 41iT^{2} \)
43 \( 1 + (1.06 + 1.06i)T + 43iT^{2} \)
47 \( 1 + (1.53 + 1.53i)T + 47iT^{2} \)
53 \( 1 + 4.74T + 53T^{2} \)
59 \( 1 - 9.85iT - 59T^{2} \)
61 \( 1 + (3.88 - 3.88i)T - 61iT^{2} \)
67 \( 1 + 1.35iT - 67T^{2} \)
71 \( 1 + 11.6iT - 71T^{2} \)
73 \( 1 + (-11.0 - 11.0i)T + 73iT^{2} \)
79 \( 1 + (-7.92 - 7.92i)T + 79iT^{2} \)
83 \( 1 + 4.45iT - 83T^{2} \)
89 \( 1 + (5.59 - 5.59i)T - 89iT^{2} \)
97 \( 1 + (-5.97 - 5.97i)T + 97iT^{2} \)
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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

−9.615740195103437770589820324740, −8.819732805387612379047030885829, −8.056262654099276168709302548267, −6.91949084590405286612235522571, −6.46446144747258869770485817451, −5.61795915195899621376712643860, −4.76003820311433980545708766939, −3.40620690409702887707785395648, −2.55700426495737915370894585624, −1.21990892834582853214620902110, 1.71078774601272322471732116611, 2.50483941172384654037353982757, 3.26367330725880274822280399209, 4.92472799414965446688471067591, 5.24244367180493637119858471264, 6.18900944972431864130894393246, 7.03197620993233271210897295998, 8.502736218066583233348046998544, 9.318164601274111715207290674610, 9.593094129957920267969818605654

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