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Properties

Label 2-1224-1224.707-c0-0-1
Degree $2$
Conductor $1224$
Sign $0.751 + 0.659i$
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  + (−0.130 − 0.991i)2-s + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)4-s + (0.130 − 0.991i)6-s + (0.382 + 0.923i)8-s + (0.866 + 0.499i)9-s + (0.793 + 0.391i)11-s − 12-s + (0.866 − 0.5i)16-s + (−0.991 + 0.130i)17-s + (0.382 − 0.923i)18-s + (0.198 − 0.478i)19-s + (0.284 − 0.837i)22-s + (0.130 + 0.991i)24-s + (0.793 + 0.608i)25-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)2-s + (0.965 + 0.258i)3-s + (−0.965 + 0.258i)4-s + (0.130 − 0.991i)6-s + (0.382 + 0.923i)8-s + (0.866 + 0.499i)9-s + (0.793 + 0.391i)11-s − 12-s + (0.866 − 0.5i)16-s + (−0.991 + 0.130i)17-s + (0.382 − 0.923i)18-s + (0.198 − 0.478i)19-s + (0.284 − 0.837i)22-s + (0.130 + 0.991i)24-s + (0.793 + 0.608i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.751 + 0.659i$
Analytic conductor: \(0.610855\)
Root analytic conductor: \(0.781572\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (707, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :0),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.298564980\)
\(L(\frac12)\) \(\approx\) \(1.298564980\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 + 0.991i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
17 \( 1 + (0.991 - 0.130i)T \)
good5 \( 1 + (-0.793 - 0.608i)T^{2} \)
7 \( 1 + (0.793 - 0.608i)T^{2} \)
11 \( 1 + (-0.793 - 0.391i)T + (0.608 + 0.793i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.198 + 0.478i)T + (-0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.991 + 0.130i)T^{2} \)
29 \( 1 + (0.130 - 0.991i)T^{2} \)
31 \( 1 + (0.608 - 0.793i)T^{2} \)
37 \( 1 + (-0.382 - 0.923i)T^{2} \)
41 \( 1 + (1.34 + 1.18i)T + (0.130 + 0.991i)T^{2} \)
43 \( 1 + (-1.20 + 1.57i)T + (-0.258 - 0.965i)T^{2} \)
47 \( 1 + (0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (1.57 + 0.207i)T + (0.965 + 0.258i)T^{2} \)
61 \( 1 + (-0.793 + 0.608i)T^{2} \)
67 \( 1 + (-0.226 - 0.130i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.382 + 0.923i)T^{2} \)
73 \( 1 + (0.128 - 0.0255i)T + (0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.608 + 0.793i)T^{2} \)
83 \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \)
89 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
97 \( 1 + (0.483 - 0.423i)T + (0.130 - 0.991i)T^{2} \)
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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

−9.724571736906910573763002469032, −9.038885630690676772697613158911, −8.661031760140904638017126119053, −7.57107840329676284962955608312, −6.75087129323119110292378060894, −5.19172597373272535439524920199, −4.33446843868588725938130210605, −3.57785108610535856607819170920, −2.55949267518326612836963443980, −1.57298079810105168248951580680, 1.40039183637644234994956365331, 2.99796886935578871110273383835, 4.05556637047602068035181097460, 4.84745023299429896256689404921, 6.25727344435638099611678703042, 6.67116760259975325371689473702, 7.66638191470057923061736113091, 8.339770190124588847262081933031, 9.021674628043241398142469811443, 9.604729471138106929850910397109

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