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Properties

Label 2-2007-223.104-c0-0-0
Degree $2$
Conductor $2007$
Sign $0.729 + 0.684i$
Analytic cond. $1.00162$
Root an. cond. $1.00081$
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  + (−0.985 − 0.169i)4-s + (1.71 − 0.776i)7-s + (−0.481 + 0.147i)13-s + (0.942 + 0.333i)16-s + (−0.386 − 0.439i)19-s + (0.873 − 0.487i)25-s + (−1.82 + 0.475i)28-s + (−0.183 + 0.175i)31-s + (−0.538 − 1.33i)37-s + (1.15 + 0.644i)43-s + (1.68 − 1.91i)49-s + (0.499 − 0.0640i)52-s + (−0.787 + 0.241i)61-s + (−0.873 − 0.487i)64-s + (0.301 − 1.76i)67-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)4-s + (1.71 − 0.776i)7-s + (−0.481 + 0.147i)13-s + (0.942 + 0.333i)16-s + (−0.386 − 0.439i)19-s + (0.873 − 0.487i)25-s + (−1.82 + 0.475i)28-s + (−0.183 + 0.175i)31-s + (−0.538 − 1.33i)37-s + (1.15 + 0.644i)43-s + (1.68 − 1.91i)49-s + (0.499 − 0.0640i)52-s + (−0.787 + 0.241i)61-s + (−0.873 − 0.487i)64-s + (0.301 − 1.76i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $0.729 + 0.684i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (550, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ 0.729 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.086177724\)
\(L(\frac12)\) \(\approx\) \(1.086177724\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 + (0.873 - 0.487i)T \)
good2 \( 1 + (0.985 + 0.169i)T^{2} \)
5 \( 1 + (-0.873 + 0.487i)T^{2} \)
7 \( 1 + (-1.71 + 0.776i)T + (0.660 - 0.750i)T^{2} \)
11 \( 1 + (0.721 + 0.691i)T^{2} \)
13 \( 1 + (0.481 - 0.147i)T + (0.828 - 0.559i)T^{2} \)
17 \( 1 + (-0.721 - 0.691i)T^{2} \)
19 \( 1 + (0.386 + 0.439i)T + (-0.127 + 0.991i)T^{2} \)
23 \( 1 + (-0.873 + 0.487i)T^{2} \)
29 \( 1 + (0.778 + 0.628i)T^{2} \)
31 \( 1 + (0.183 - 0.175i)T + (0.0424 - 0.999i)T^{2} \)
37 \( 1 + (0.538 + 1.33i)T + (-0.721 + 0.691i)T^{2} \)
41 \( 1 + (-0.292 + 0.956i)T^{2} \)
43 \( 1 + (-1.15 - 0.644i)T + (0.524 + 0.851i)T^{2} \)
47 \( 1 + (-0.127 - 0.991i)T^{2} \)
53 \( 1 + (-0.911 - 0.411i)T^{2} \)
59 \( 1 + (0.594 - 0.803i)T^{2} \)
61 \( 1 + (0.787 - 0.241i)T + (0.828 - 0.559i)T^{2} \)
67 \( 1 + (-0.301 + 1.76i)T + (-0.942 - 0.333i)T^{2} \)
71 \( 1 + (0.594 - 0.803i)T^{2} \)
73 \( 1 + (-0.250 - 1.95i)T + (-0.967 + 0.251i)T^{2} \)
79 \( 1 + (-0.784 + 0.579i)T + (0.292 - 0.956i)T^{2} \)
83 \( 1 + (0.524 - 0.851i)T^{2} \)
89 \( 1 + (0.873 + 0.487i)T^{2} \)
97 \( 1 + (-0.405 + 1.55i)T + (-0.873 - 0.487i)T^{2} \)
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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

−9.082732791285321203186462669628, −8.532996626885514690358649934122, −7.75312811071546854204355316427, −7.14444432690146034917723229995, −5.90546038506127819978667515643, −4.87093181876810966611832723051, −4.62250007408118678806876260914, −3.68812870085335233444155623771, −2.16212764764612032252591686251, −0.957921428897014067004852950262, 1.38262654032254869950646932586, 2.55132967108389766066190531976, 3.80471037783736353580242416576, 4.82610725196068100185753377325, 5.11851132196001626207891362755, 6.06675928361181729444281317135, 7.40374579658185585983062696709, 8.005939727947818898594050433137, 8.695333115349240942020146745177, 9.152717047070456443626789064207

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