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Properties

Label 2-1001-7.2-c1-0-67
Degree $2$
Conductor $1001$
Sign $-0.188 + 0.982i$
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  + (1.07 − 1.87i)2-s + (0.921 + 1.59i)3-s + (−1.33 − 2.30i)4-s + (0.721 − 1.25i)5-s + 3.97·6-s + (1.85 − 1.88i)7-s − 1.43·8-s + (−0.197 + 0.341i)9-s + (−1.55 − 2.70i)10-s + (−0.5 − 0.866i)11-s + (2.45 − 4.24i)12-s − 13-s + (−1.52 − 5.50i)14-s + 2.66·15-s + (1.11 − 1.93i)16-s + (−1.16 − 2.01i)17-s + ⋯
L(s)  = 1  + (0.763 − 1.32i)2-s + (0.531 + 0.921i)3-s + (−0.665 − 1.15i)4-s + (0.322 − 0.559i)5-s + 1.62·6-s + (0.700 − 0.713i)7-s − 0.506·8-s + (−0.0657 + 0.113i)9-s + (−0.492 − 0.853i)10-s + (−0.150 − 0.261i)11-s + (0.708 − 1.22i)12-s − 0.277·13-s + (−0.408 − 1.47i)14-s + 0.686·15-s + (0.279 − 0.483i)16-s + (−0.281 − 0.487i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.188 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1001\)    =    \(7 \cdot 11 \cdot 13\)
Sign: $-0.188 + 0.982i$
Analytic conductor: \(7.99302\)
Root analytic conductor: \(2.82719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1001} (716, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1001,\ (\ :1/2),\ -0.188 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.181792939\)
\(L(\frac12)\) \(\approx\) \(3.181792939\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.85 + 1.88i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
good2 \( 1 + (-1.07 + 1.87i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.921 - 1.59i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.721 + 1.25i)T + (-2.5 - 4.33i)T^{2} \)
17 \( 1 + (1.16 + 2.01i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.92 - 5.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.78 - 4.82i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.21T + 29T^{2} \)
31 \( 1 + (3.52 + 6.10i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.97 + 3.42i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2.86T + 41T^{2} \)
43 \( 1 - 9.67T + 43T^{2} \)
47 \( 1 + (4.03 - 6.98i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.0397 + 0.0688i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.63 - 4.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.84 + 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + (-4.50 - 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.50 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (-0.730 + 1.26i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.90T + 97T^{2} \)
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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

−9.816784230626714962677424299819, −9.405039666258699931066339526359, −8.303545811312993968853851182728, −7.40348428822808660281257201347, −5.79856453641901506095900622971, −4.89420415045849170184066629556, −4.16881112396143601311729944768, −3.59674920356972643958364504604, −2.36834059914408706375155593551, −1.21605502941520564668734650093, 1.91380098420981264284975795553, 2.76563452558941859897893167753, 4.41733814596777335167587246349, 5.08786424669712473572435098973, 6.23216902415840246949824890308, 6.75203362533836755131827882740, 7.46525270735970227085333277776, 8.371730193807544684473993216581, 8.716579387815386009151227642309, 10.24708057985147908631844994191

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