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Properties

Label 2-301-301.100-c1-0-10
Degree $2$
Conductor $301$
Sign $0.0845 - 0.996i$
Analytic cond. $2.40349$
Root an. cond. $1.55032$
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  + (−0.0880 + 1.17i)2-s + (0.943 − 0.643i)3-s + (0.606 + 0.0913i)4-s + (−0.622 + 2.72i)5-s + (0.672 + 1.16i)6-s + (1.05 − 2.42i)7-s + (−0.684 + 3.00i)8-s + (−0.619 + 1.57i)9-s + (−3.14 − 0.971i)10-s + (0.0805 − 0.205i)11-s + (0.630 − 0.303i)12-s + (0.907 + 0.842i)13-s + (2.75 + 1.45i)14-s + (1.16 + 2.97i)15-s + (−2.29 − 0.707i)16-s + (−0.669 − 2.93i)17-s + ⋯
L(s)  = 1  + (−0.0622 + 0.830i)2-s + (0.544 − 0.371i)3-s + (0.303 + 0.0456i)4-s + (−0.278 + 1.21i)5-s + (0.274 + 0.475i)6-s + (0.399 − 0.916i)7-s + (−0.242 + 1.06i)8-s + (−0.206 + 0.525i)9-s + (−0.995 − 0.307i)10-s + (0.0242 − 0.0619i)11-s + (0.182 − 0.0876i)12-s + (0.251 + 0.233i)13-s + (0.736 + 0.388i)14-s + (0.301 + 0.768i)15-s + (−0.573 − 0.176i)16-s + (−0.162 − 0.711i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0845 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0845 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(301\)    =    \(7 \cdot 43\)
Sign: $0.0845 - 0.996i$
Analytic conductor: \(2.40349\)
Root analytic conductor: \(1.55032\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{301} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 301,\ (\ :1/2),\ 0.0845 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20176 + 1.10411i\)
\(L(\frac12)\) \(\approx\) \(1.20176 + 1.10411i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.05 + 2.42i)T \)
43 \( 1 + (-1.86 + 6.28i)T \)
good2 \( 1 + (0.0880 - 1.17i)T + (-1.97 - 0.298i)T^{2} \)
3 \( 1 + (-0.943 + 0.643i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (0.622 - 2.72i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (-0.0805 + 0.205i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-0.907 - 0.842i)T + (0.971 + 12.9i)T^{2} \)
17 \( 1 + (0.669 + 2.93i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (-1.43 + 1.80i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (-1.80 - 2.26i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.136 + 1.82i)T + (-28.6 - 4.32i)T^{2} \)
31 \( 1 + (-0.484 + 6.46i)T + (-30.6 - 4.62i)T^{2} \)
37 \( 1 + 2.79T + 37T^{2} \)
41 \( 1 + (3.71 - 1.79i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (1.17 + 2.98i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (4.22 + 1.30i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (-8.65 + 8.03i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (0.111 + 1.48i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (-5.63 - 14.3i)T + (-49.1 + 45.5i)T^{2} \)
71 \( 1 + (1.26 + 3.21i)T + (-52.0 + 48.2i)T^{2} \)
73 \( 1 + (-2.59 + 11.3i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.141 - 0.245i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.28 + 17.1i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-0.387 - 0.186i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-1.77 - 2.22i)T + (-21.5 + 94.5i)T^{2} \)
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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

−11.50799843611247105153173225342, −11.22862598703413223618808072740, −10.20596559980889000392144223283, −8.738618750369207941350300387232, −7.68650789755167457693413984053, −7.28879857846755262731052999917, −6.49660688709737097123155920895, −5.08664418143894856670324882347, −3.41024871995876177471800804560, −2.24228315694674553825551533160, 1.36260622171389249414386577423, 2.86186920268601404128357191492, 4.00806135575667367327588684610, 5.28094266596722749670221460069, 6.51096606380297804186881110440, 8.166776624602697301598251960018, 8.807924319781712324086813311723, 9.556480211390430031526075952608, 10.63907925155972405108898261388, 11.65565367803945226052867153837

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