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Properties

Label 2-1950-65.49-c1-0-13
Degree $2$
Conductor $1950$
Sign $0.838 + 0.545i$
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  + (−0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (0.866 + 0.499i)6-s + (−0.758 + 1.31i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−4.45 + 2.57i)11-s − 0.999i·12-s + (−1.86 − 3.08i)13-s + 1.51·14-s + (−0.5 − 0.866i)16-s + (−4.15 − 2.39i)17-s − 0.999·18-s + (1.39 + 0.807i)19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.353 + 0.204i)6-s + (−0.286 + 0.496i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−1.34 + 0.775i)11-s − 0.288i·12-s + (−0.516 − 0.856i)13-s + 0.405·14-s + (−0.125 − 0.216i)16-s + (−1.00 − 0.581i)17-s − 0.235·18-s + (0.320 + 0.185i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.838 + 0.545i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.838 + 0.545i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7805028789\)
\(L(\frac12)\) \(\approx\) \(0.7805028789\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (1.86 + 3.08i)T \)
good7 \( 1 + (0.758 - 1.31i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.45 - 2.57i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (4.15 + 2.39i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.39 - 0.807i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.71 + 2.72i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.93 - 8.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 9.88iT - 31T^{2} \)
37 \( 1 + (4.36 + 7.56i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.10 + 0.637i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.04 + 1.18i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 6.84T + 47T^{2} \)
53 \( 1 - 0.881iT - 53T^{2} \)
59 \( 1 + (-8.04 - 4.64i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.24 - 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.76 - 9.98i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-13.7 - 7.91i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 4.39T + 73T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + 0.979T + 83T^{2} \)
89 \( 1 + (-3.54 + 2.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.85 + 4.95i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.218361511580727723607840899879, −8.565853223091701499619422963111, −7.49654405203049419445021607952, −6.97890954444811112689507906428, −5.64232793944466257582722048329, −5.08918758500530027913108530545, −4.22426481360524093261888289594, −2.88031956772704026926351068217, −2.34325469406080357286548435977, −0.56306300588887861210989723122, 0.67837799806466063132188279196, 2.15683525603600089974425401368, 3.42238329170287864056191686591, 4.75709411137341057045669152569, 5.20340579482649548376855765979, 6.41345615320794306612146585095, 6.73873287035463676797661617995, 7.67645699872512769601541221472, 8.334059144977820391634140840232, 9.150704171811977189078861322654

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