Properties

Label 2-950-5.4-c1-0-22
Degree $2$
Conductor $950$
Sign $-0.894 + 0.447i$
Analytic cond. $7.58578$
Root an. cond. $2.75423$
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 i·3-s − 4-s − 6-s − 3i·7-s + i·8-s + 2·9-s + 2·11-s + i·12-s i·13-s − 3·14-s + 16-s − 3i·17-s − 2i·18-s + 19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.13i·7-s + 0.353i·8-s + 0.666·9-s + 0.603·11-s + 0.288i·12-s − 0.277i·13-s − 0.801·14-s + 0.250·16-s − 0.727i·17-s − 0.471i·18-s + 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342341 - 1.45018i\)
\(L(\frac12)\) \(\approx\) \(0.342341 - 1.45018i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 + 15T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + 3iT - 67T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−9.896540214040990582267120060212, −8.993274606774424378511424752687, −7.919176603635043067459351098544, −7.18071432601313327592947201680, −6.47540811791126218630180644646, −5.06327090117280067815267076119, −4.16604537503819153879657329221, −3.24569016120247373212380306101, −1.79230948075124349973494254521, −0.75062600675756926202476192265, 1.73532994630650044859878851120, 3.35882232367253671756552109964, 4.33044112260669478401994054540, 5.22989644326966415173770998739, 6.08546600481128978129866741152, 6.92435317740451826562152346022, 7.908104386178803227334811603934, 8.921527482239301520337262919554, 9.290777812594767207445269516254, 10.20686177070777993702787758356

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