Properties

Label 2-950-19.17-c1-0-29
Degree $2$
Conductor $950$
Sign $-0.997 - 0.0646i$
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  + (0.173 − 0.984i)2-s + (−0.0864 − 0.0725i)3-s + (−0.939 − 0.342i)4-s + (−0.0864 + 0.0725i)6-s + (−0.772 − 1.33i)7-s + (−0.5 + 0.866i)8-s + (−0.518 − 2.94i)9-s + (0.653 − 1.13i)11-s + (0.0564 + 0.0977i)12-s + (0.437 − 0.367i)13-s + (−1.45 + 0.528i)14-s + (0.766 + 0.642i)16-s + (−0.878 + 4.98i)17-s − 2.98·18-s + (−0.546 − 4.32i)19-s + ⋯
L(s)  = 1  + (0.122 − 0.696i)2-s + (−0.0499 − 0.0418i)3-s + (−0.469 − 0.171i)4-s + (−0.0353 + 0.0296i)6-s + (−0.291 − 0.505i)7-s + (−0.176 + 0.306i)8-s + (−0.172 − 0.980i)9-s + (0.196 − 0.341i)11-s + (0.0162 + 0.0282i)12-s + (0.121 − 0.101i)13-s + (−0.387 + 0.141i)14-s + (0.191 + 0.160i)16-s + (−0.212 + 1.20i)17-s − 0.704·18-s + (−0.125 − 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(950\)    =    \(2 \cdot 5^{2} \cdot 19\)
Sign: $-0.997 - 0.0646i$
Analytic conductor: \(7.58578\)
Root analytic conductor: \(2.75423\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{950} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 950,\ (\ :1/2),\ -0.997 - 0.0646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0280689 + 0.866881i\)
\(L(\frac12)\) \(\approx\) \(0.0280689 + 0.866881i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.173 + 0.984i)T \)
5 \( 1 \)
19 \( 1 + (0.546 + 4.32i)T \)
good3 \( 1 + (0.0864 + 0.0725i)T + (0.520 + 2.95i)T^{2} \)
7 \( 1 + (0.772 + 1.33i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.653 + 1.13i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.437 + 0.367i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (0.878 - 4.98i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (4.86 + 1.77i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.660 - 3.74i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (1.49 + 2.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.31T + 37T^{2} \)
41 \( 1 + (8.57 + 7.19i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (8.85 - 3.22i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.368 + 2.08i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (7.19 + 2.61i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (0.0463 - 0.262i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-3.05 - 1.11i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (1.57 + 8.94i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-15.2 + 5.55i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (2.14 + 1.80i)T + (12.6 + 71.8i)T^{2} \)
79 \( 1 + (-3.19 - 2.68i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-4.63 - 8.03i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.42 - 3.71i)T + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (0.500 - 2.83i)T + (-91.1 - 33.1i)T^{2} \)
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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

−9.756351469395601670806095380990, −8.862473859682153601095688631965, −8.216614558777538263931735561505, −6.83835104918606751431472927792, −6.25144245962541601598896043110, −5.10119267108084717007782194332, −3.89455988961314902102453398860, −3.35260363939921756883465962125, −1.86323160079845992221625632130, −0.37849328032389022972868291267, 1.96388746206689266718906000461, 3.28223517836354316199695675047, 4.51873766840248590208686943223, 5.31829086249061033958201782338, 6.15778927313165249470195919110, 7.07338431034392659592481197941, 7.952540726733756136993048672429, 8.607438144333366503170542936297, 9.642058875146193505443334291740, 10.19576779305194283131815987685

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