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Properties

Label 2-301-301.100-c1-0-9
Degree $2$
Conductor $301$
Sign $-0.617 - 0.786i$
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  + (−0.136 + 1.81i)2-s + (−0.580 + 0.395i)3-s + (−1.29 − 0.195i)4-s + (0.502 − 2.20i)5-s + (−0.639 − 1.10i)6-s + (2.64 − 0.0762i)7-s + (−0.277 + 1.21i)8-s + (−0.915 + 2.33i)9-s + (3.92 + 1.21i)10-s + (−1.52 + 3.87i)11-s + (0.831 − 0.400i)12-s + (1.31 + 1.22i)13-s + (−0.221 + 4.81i)14-s + (0.579 + 1.47i)15-s + (−4.68 − 1.44i)16-s + (1.16 + 5.09i)17-s + ⋯
L(s)  = 1  + (−0.0962 + 1.28i)2-s + (−0.335 + 0.228i)3-s + (−0.649 − 0.0979i)4-s + (0.224 − 0.983i)5-s + (−0.261 − 0.452i)6-s + (0.999 − 0.0288i)7-s + (−0.0982 + 0.430i)8-s + (−0.305 + 0.777i)9-s + (1.24 + 0.382i)10-s + (−0.458 + 1.16i)11-s + (0.240 − 0.115i)12-s + (0.365 + 0.338i)13-s + (−0.0591 + 1.28i)14-s + (0.149 + 0.381i)15-s + (−1.17 − 0.361i)16-s + (0.282 + 1.23i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(301\)    =    \(7 \cdot 43\)
Sign: $-0.617 - 0.786i$
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: \((2,\ 301,\ (\ :1/2),\ -0.617 - 0.786i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.536204 + 1.10309i\)
\(L(\frac12)\) \(\approx\) \(0.536204 + 1.10309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.64 + 0.0762i)T \)
43 \( 1 + (6.55 - 0.151i)T \)
good2 \( 1 + (0.136 - 1.81i)T + (-1.97 - 0.298i)T^{2} \)
3 \( 1 + (0.580 - 0.395i)T + (1.09 - 2.79i)T^{2} \)
5 \( 1 + (-0.502 + 2.20i)T + (-4.50 - 2.16i)T^{2} \)
11 \( 1 + (1.52 - 3.87i)T + (-8.06 - 7.48i)T^{2} \)
13 \( 1 + (-1.31 - 1.22i)T + (0.971 + 12.9i)T^{2} \)
17 \( 1 + (-1.16 - 5.09i)T + (-15.3 + 7.37i)T^{2} \)
19 \( 1 + (-3.18 + 3.99i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (0.501 + 0.629i)T + (-5.11 + 22.4i)T^{2} \)
29 \( 1 + (-0.536 + 7.16i)T + (-28.6 - 4.32i)T^{2} \)
31 \( 1 + (0.358 - 4.78i)T + (-30.6 - 4.62i)T^{2} \)
37 \( 1 - 5.28T + 37T^{2} \)
41 \( 1 + (2.26 - 1.09i)T + (25.5 - 32.0i)T^{2} \)
47 \( 1 + (2.64 + 6.73i)T + (-34.4 + 31.9i)T^{2} \)
53 \( 1 + (-4.93 - 1.52i)T + (43.7 + 29.8i)T^{2} \)
59 \( 1 + (3.47 - 3.21i)T + (4.40 - 58.8i)T^{2} \)
61 \( 1 + (0.242 + 3.23i)T + (-60.3 + 9.09i)T^{2} \)
67 \( 1 + (3.86 + 9.84i)T + (-49.1 + 45.5i)T^{2} \)
71 \( 1 + (-3.57 - 9.11i)T + (-52.0 + 48.2i)T^{2} \)
73 \( 1 + (-3.18 + 13.9i)T + (-65.7 - 31.6i)T^{2} \)
79 \( 1 + (2.55 + 4.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.296 + 3.95i)T + (-82.0 + 12.3i)T^{2} \)
89 \( 1 + (-5.75 - 2.77i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (0.481 + 0.604i)T + (-21.5 + 94.5i)T^{2} \)
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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

−12.01314289456404818571867755324, −11.17232297224867468815747763648, −10.08169474178894100623130071498, −8.828025604584223324874622870566, −8.144410705072109061944376246482, −7.37539588318311819979405563051, −6.05573806111797823535594758442, −5.03345058619210376329214955141, −4.65679097746289678238717436796, −1.96429238186065609597865009690, 1.07340194991767245776665739207, 2.75437082060504855241630968616, 3.54413192750635677717614830090, 5.35584659348650697426912745364, 6.40035449857669859217919260330, 7.57157073975993707703852371088, 8.798246256317332858306936497120, 9.904161861354768633384260221482, 10.76530361257030082183791348969, 11.41537164223268332807958037692

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