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Properties

Label 2-1001-7.4-c1-0-54
Degree $2$
Conductor $1001$
Sign $-0.921 - 0.388i$
Analytic cond. $7.99302$
Root an. cond. $2.82719$
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  + (−0.838 − 1.45i)2-s + (−1.35 + 2.35i)3-s + (−0.406 + 0.704i)4-s + (−0.392 − 0.680i)5-s + 4.55·6-s + (2.59 − 0.494i)7-s − 1.99·8-s + (−2.18 − 3.78i)9-s + (−0.658 + 1.14i)10-s + (−0.5 + 0.866i)11-s + (−1.10 − 1.91i)12-s − 13-s + (−2.89 − 3.36i)14-s + 2.13·15-s + (2.48 + 4.29i)16-s + (−2.26 + 3.93i)17-s + ⋯
L(s)  = 1  + (−0.593 − 1.02i)2-s + (−0.783 + 1.35i)3-s + (−0.203 + 0.352i)4-s + (−0.175 − 0.304i)5-s + 1.85·6-s + (0.982 − 0.186i)7-s − 0.703·8-s + (−0.728 − 1.26i)9-s + (−0.208 + 0.360i)10-s + (−0.150 + 0.261i)11-s + (−0.318 − 0.551i)12-s − 0.277·13-s + (−0.774 − 0.898i)14-s + 0.550·15-s + (0.620 + 1.07i)16-s + (−0.550 + 0.953i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 - 0.388i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.921 - 0.388i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ -0.921 - 0.388i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02202715889\)
\(L(\frac12)\) \(\approx\) \(0.02202715889\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.59 + 0.494i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.838 + 1.45i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (1.35 - 2.35i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.392 + 0.680i)T + (-2.5 + 4.33i)T^{2} \)
17 \( 1 + (2.26 - 3.93i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.33 + 4.03i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.112 + 0.194i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.96T + 29T^{2} \)
31 \( 1 + (2.50 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.217 - 0.376i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.9T + 41T^{2} \)
43 \( 1 + 1.85T + 43T^{2} \)
47 \( 1 + (5.32 + 9.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.21 - 3.82i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.85 - 11.8i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.41 + 9.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.73 - 8.20i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.6T + 71T^{2} \)
73 \( 1 + (5.49 - 9.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.51 + 9.55i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (3.83 + 6.64i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 13.0T + 97T^{2} \)
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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

−9.948334727766996926990076501290, −8.670635145349483594347622929868, −8.580742999155229487339531754691, −6.89697804342836957765143209249, −5.81132025295599245552494820273, −4.74814053152951718064817990641, −4.34693673983269707632890343682, −3.00719464089429891146185704838, −1.61737878096455796204400327195, −0.01365629346505705435601109590, 1.56048534086887369991809964036, 2.90108574154803119829057903929, 4.78492817156829957115750063576, 5.69153260168828648233947292315, 6.47175397458900685146149353555, 7.09632411819890257718808197851, 7.82869974673981727462225404141, 8.295117081234141552073668907366, 9.281452098603492357954291183373, 10.54492507467530042161430632217

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