Properties

Label 2-1950-65.64-c1-0-36
Degree $2$
Conductor $1950$
Sign $-0.581 + 0.813i$
Analytic cond. $15.5708$
Root an. cond. $3.94598$
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  − 2-s i·3-s + 4-s + i·6-s + 3.12·7-s − 8-s − 9-s − 5.12i·11-s i·12-s + (0.561 − 3.56i)13-s − 3.12·14-s + 16-s + 2i·17-s + 18-s − 6i·19-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577i·3-s + 0.5·4-s + 0.408i·6-s + 1.18·7-s − 0.353·8-s − 0.333·9-s − 1.54i·11-s − 0.288i·12-s + (0.155 − 0.987i)13-s − 0.834·14-s + 0.250·16-s + 0.485i·17-s + 0.235·18-s − 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.581 + 0.813i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ -0.581 + 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.180151944\)
\(L(\frac12)\) \(\approx\) \(1.180151944\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (-0.561 + 3.56i)T \)
good7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 5.12iT - 11T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 - 5.12iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 3.12iT - 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 + 0.876iT - 41T^{2} \)
43 \( 1 + 6.24iT - 43T^{2} \)
47 \( 1 + 6.24T + 47T^{2} \)
53 \( 1 - 13.3iT - 53T^{2} \)
59 \( 1 + 1.12iT - 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 4.87T + 67T^{2} \)
71 \( 1 + 10.2iT - 71T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6.24T + 83T^{2} \)
89 \( 1 - 3.12iT - 89T^{2} \)
97 \( 1 + 13.1T + 97T^{2} \)
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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

−8.678677933958750871834344566111, −8.176664550372164133919940557089, −7.60649812270341498530079491582, −6.69910724033158905015173591054, −5.72622434175477454554832794838, −5.18417027208865692373149661992, −3.69960303142110675603502089834, −2.72749167829399010920066429819, −1.56846233603366561074380627698, −0.55184023070472585412947222210, 1.54247966423304117161995704142, 2.26272141564055991122881952317, 3.76299799816882694576718587296, 4.63463139422776371304037847504, 5.26028132892210687954445313447, 6.53178135869764567401411835335, 7.18816333573932732179343715331, 8.145057629330969187166984129762, 8.574426917416526637114467052008, 9.685887977641058566095011105584

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