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Properties

Label 2-350-7.2-c1-0-1
Degree $2$
Conductor $350$
Sign $-0.701 - 0.712i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (−0.5 + 0.866i)2-s + (1.5 + 2.59i)3-s + (−0.499 − 0.866i)4-s − 3·6-s + (2.5 + 0.866i)7-s + 0.999·8-s + (−3 + 5.19i)9-s + (1.50 − 2.59i)12-s − 2·13-s + (−2 + 1.73i)14-s + (−0.5 + 0.866i)16-s + (1 + 1.73i)17-s + (−3 − 5.19i)18-s + (1 − 1.73i)19-s + (1.5 + 7.79i)21-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.866 + 1.49i)3-s + (−0.249 − 0.433i)4-s − 1.22·6-s + (0.944 + 0.327i)7-s + 0.353·8-s + (−1 + 1.73i)9-s + (0.433 − 0.749i)12-s − 0.554·13-s + (−0.534 + 0.462i)14-s + (−0.125 + 0.216i)16-s + (0.242 + 0.420i)17-s + (−0.707 − 1.22i)18-s + (0.229 − 0.397i)19-s + (0.327 + 1.70i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.701 - 0.712i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.701 - 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.577166 + 1.37721i\)
\(L(\frac12)\) \(\approx\) \(0.577166 + 1.37721i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good3 \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.5 + 7.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5 + 8.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.39672349794215585897234203064, −10.70579498926116753181838898832, −9.676889340828289790044672201961, −9.134326316220532963709506001841, −8.219878552554470314574620495250, −7.49981925385182309215577814242, −5.74640849959557748866390971630, −4.84614784415942484536992430751, −3.93668841716346014200728726322, −2.37431024570718377028093746445, 1.20332184231598067930233316127, 2.25996547662166844399380393455, 3.49151910112258463507541597741, 5.13347973556507337405944809717, 6.79608033597640837649799375189, 7.56765317661854420775095178722, 8.238153075996555691246610215813, 9.072981499623146128113275548746, 10.20388157130398291704756193339, 11.40069568547317101548102551534

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