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Properties

Label 2-350-7.2-c1-0-0
Degree $2$
Conductor $350$
Sign $0.266 - 0.963i$
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 + (−0.499 − 0.866i)4-s + (0.5 + 2.59i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (1 + 1.73i)11-s + (−2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (3.5 + 6.06i)17-s + (1.5 + 2.59i)18-s − 1.99·22-s + (−1.5 + 2.59i)23-s + (2 − 1.73i)28-s + 6·29-s + (3.5 + 6.06i)31-s + (−0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.188 + 0.981i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.301 + 0.522i)11-s + (−0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.848 + 1.47i)17-s + (0.353 + 0.612i)18-s − 0.426·22-s + (−0.312 + 0.541i)23-s + (0.377 − 0.327i)28-s + 1.11·29-s + (0.628 + 1.08i)31-s + (−0.0883 − 0.153i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.266 - 0.963i$
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: \((2,\ 350,\ (\ :1/2),\ 0.266 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917926 + 0.698317i\)
\(L(\frac12)\) \(\approx\) \(0.917926 + 0.698317i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good3 \( 1 + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-3.5 - 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 7T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.5 + 6.06i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7 - 12.1i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7 + 12.1i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + T + 71T^{2} \)
73 \( 1 + (7 + 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 14T + 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−11.97622288247141628084182420243, −10.45231566619500019162499392638, −9.742630520186105559131812883250, −8.786507306431124316737188132667, −8.047525054642078981280257566202, −6.80375593339708622047049576720, −6.04801085419458992481377128659, −4.92439453647826592655528096921, −3.55266354013516752286358624908, −1.62687141114993845125987658539, 1.05374713058931226300627199620, 2.75151428443267492863522121346, 4.11661633352941495922783940720, 5.08733248656995524803886834053, 6.72731984865423189383662011201, 7.68511200006167081488430338546, 8.441731629253052977476993093387, 9.829390911637106665841894047534, 10.24064832687969335178722925582, 11.29685536882335947917166577368

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