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Properties

Label 2-350-5.4-c1-0-0
Degree $2$
Conductor $350$
Sign $-0.894 + 0.447i$
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  + i·2-s + 3i·3-s − 4-s − 3·6-s − i·7-s − i·8-s − 6·9-s − 5·11-s − 3i·12-s + 6i·13-s + 14-s + 16-s − i·17-s − 6i·18-s + 3·19-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.73i·3-s − 0.5·4-s − 1.22·6-s − 0.377i·7-s − 0.353i·8-s − 2·9-s − 1.50·11-s − 0.866i·12-s + 1.66i·13-s + 0.267·14-s + 0.250·16-s − 0.242i·17-s − 1.41i·18-s + 0.688·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210675 - 0.892434i\)
\(L(\frac12)\) \(\approx\) \(0.210675 - 0.892434i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good3 \( 1 - 3iT - 3T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 + iT - 17T^{2} \)
19 \( 1 - 3T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 8iT - 37T^{2} \)
41 \( 1 - 11T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 + 4iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 9iT - 67T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 - 7iT - 73T^{2} \)
79 \( 1 - 2T + 79T^{2} \)
83 \( 1 + 11iT - 83T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−11.71213532525521739404529331675, −10.85847408081630898113138468309, −9.994343734392532640515735584001, −9.370667495287345675425294175021, −8.432901335985605660532466935114, −7.34160332609263357086643054592, −6.00923260737645566883775136857, −4.89340268853846218069727323280, −4.35538657252140008829443741148, −3.01173303664762003186045602547, 0.62531841110652152098686894891, 2.25592963732092920140091820731, 3.09017020285620483598189848717, 5.26795596119065845092726418093, 5.95850393584290940415092698715, 7.51444651436689669043190127183, 7.900121616147704622314811869964, 8.921340809713994409888151596341, 10.35013153478523174428802262470, 11.03248914126200209180492012179

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