Properties

Label 2-1224-408.389-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.513 + 0.857i$
Analytic cond. $0.610855$
Root an. cond. $0.781572$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.70 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (1.70 − 0.707i)14-s − 1.00·16-s + i·17-s + (−0.707 − 0.292i)23-s + (−0.707 + 0.707i)25-s + (0.707 − 1.70i)28-s + (−0.707 − 1.70i)31-s + (−0.707 + 0.707i)32-s + (0.707 + 0.707i)34-s + (0.707 − 1.70i)41-s + (−0.707 + 0.292i)46-s − 1.41·47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (1.70 + 0.707i)7-s + (−0.707 − 0.707i)8-s + (1.70 − 0.707i)14-s − 1.00·16-s + i·17-s + (−0.707 − 0.292i)23-s + (−0.707 + 0.707i)25-s + (0.707 − 1.70i)28-s + (−0.707 − 1.70i)31-s + (−0.707 + 0.707i)32-s + (0.707 + 0.707i)34-s + (0.707 − 1.70i)41-s + (−0.707 + 0.292i)46-s − 1.41·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.513 + 0.857i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (1205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.513 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.702483346\)
\(L(\frac12)\) \(\approx\) \(1.702483346\)
\(L(1)\) 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 \( 1 \)
17 \( 1 - iT \)
good5 \( 1 + (0.707 - 0.707i)T^{2} \)
7 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (0.707 - 0.707i)T^{2} \)
41 \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - iT^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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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

−9.963759920739648523989276983339, −9.032273034779138521086367405805, −8.249889994833035624905979481068, −7.39769421723455263637070589524, −5.97328851808963934472003935638, −5.53690399569354847245813687953, −4.52707859472229565265934182160, −3.79899646007313842851145514025, −2.31128585095778361805872346941, −1.64611876334853227501194092828, 1.74957326672877582174375742629, 3.16315848821509543028962822456, 4.37155640023847310752728970797, 4.82842798297936464632734958550, 5.74520584777506317499909352236, 6.82370450875603030219720714401, 7.62744963743802010497337797874, 8.094289855583212984763297607195, 8.955779551721176289336551690066, 10.09853589921631704620052454385

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