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Properties

Label 2-1003-1003.648-c0-0-1
Degree $2$
Conductor $1003$
Sign $0.739 - 0.673i$
Analytic cond. $0.500562$
Root an. cond. $0.707504$
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  + (1.12 − 0.465i)3-s + i·4-s + (−0.0999 − 0.241i)5-s + (−0.607 + 1.46i)7-s + (0.341 − 0.341i)9-s + (0.465 + 1.12i)12-s + (−0.224 − 0.224i)15-s − 16-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)19-s + (0.241 − 0.0999i)20-s + 1.93i·21-s + (0.658 − 0.658i)25-s + (−0.241 + 0.582i)27-s + (−1.46 − 0.607i)28-s + (−0.465 − 1.12i)29-s + ⋯
L(s)  = 1  + (1.12 − 0.465i)3-s + i·4-s + (−0.0999 − 0.241i)5-s + (−0.607 + 1.46i)7-s + (0.341 − 0.341i)9-s + (0.465 + 1.12i)12-s + (−0.224 − 0.224i)15-s − 16-s + (0.707 − 0.707i)17-s + (0.707 + 0.707i)19-s + (0.241 − 0.0999i)20-s + 1.93i·21-s + (0.658 − 0.658i)25-s + (−0.241 + 0.582i)27-s + (−1.46 − 0.607i)28-s + (−0.465 − 1.12i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1003\)    =    \(17 \cdot 59\)
Sign: $0.739 - 0.673i$
Analytic conductor: \(0.500562\)
Root analytic conductor: \(0.707504\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1003} (648, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1003,\ (\ :0),\ 0.739 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.332393544\)
\(L(\frac12)\) \(\approx\) \(1.332393544\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 + 0.707i)T \)
good2 \( 1 - iT^{2} \)
3 \( 1 + (-1.12 + 0.465i)T + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.0999 + 0.241i)T + (-0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.607 - 1.46i)T + (-0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + T^{2} \)
19 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.465 + 1.12i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.707 + 0.707i)T^{2} \)
37 \( 1 + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.0999 + 0.241i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
61 \( 1 + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (1.83 + 0.758i)T + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.707 - 0.707i)T^{2} \)
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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

−9.828513055819595407171065460676, −9.230857563662291723227343952443, −8.496418119754159005384381973779, −7.985255010083625687479846297314, −7.20180996617331680831069501536, −6.10052480296346528660501352599, −5.04177756439518457701337583630, −3.60384124086231015910307526542, −2.91875205153366249179164225574, −2.14175756377514327769747975781, 1.27293537117150382696958656217, 2.92714621327352683400653285945, 3.68395362707258442720718035022, 4.62673586418386036347261117615, 5.78929304210680759859543346959, 6.88927375547154167343491511026, 7.46530159203049109349038518785, 8.605956102417916331923467213766, 9.445841793249156941886937640375, 9.949728972081556901502819962510

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