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Properties

Label 2-1950-65.49-c1-0-2
Degree $2$
Conductor $1950$
Sign $-0.891 + 0.452i$
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 + (−1.79 + 3.10i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−4.72 + 2.72i)11-s + 0.999i·12-s + (3.60 − 0.0664i)13-s − 3.59·14-s + (−0.5 − 0.866i)16-s + (−6.33 − 3.65i)17-s + 0.999·18-s + (3.34 + 1.93i)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 + (−0.678 + 1.17i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−1.42 + 0.821i)11-s + 0.288i·12-s + (0.999 − 0.0184i)13-s − 0.959·14-s + (−0.125 − 0.216i)16-s + (−1.53 − 0.886i)17-s + 0.235·18-s + (0.767 + 0.443i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.891 + 0.452i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.891 + 0.452i$
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: \((2,\ 1950,\ (\ :1/2),\ -0.891 + 0.452i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6289056083\)
\(L(\frac12)\) \(\approx\) \(0.6289056083\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.60 + 0.0664i)T \)
good7 \( 1 + (1.79 - 3.10i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (4.72 - 2.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (6.33 + 3.65i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.34 - 1.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.60 - 0.929i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.67 + 4.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.23iT - 31T^{2} \)
37 \( 1 + (1.06 + 1.84i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.33 - 3.65i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (7.10 + 4.10i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 5.07T + 47T^{2} \)
53 \( 1 + 2.23iT - 53T^{2} \)
59 \( 1 + (-0.237 - 0.137i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.06 + 3.57i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.39 - 12.8i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-12.3 - 7.14i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 14.8T + 73T^{2} \)
79 \( 1 + 2.62T + 79T^{2} \)
83 \( 1 + 7.15T + 83T^{2} \)
89 \( 1 + (8.30 - 4.79i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.954 + 1.65i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.509664947656769586755755571015, −8.644411287214936694501481722344, −8.122983393886563789391788767168, −7.21729717321795434400581991402, −6.52474833559660226192814877959, −5.65543914392856053774719257635, −4.99190545406576599419831427978, −3.83567817587410713613783929120, −2.82090547679344572145392613780, −2.10999524456376934079463283747, 0.17265401816126950471369345619, 1.68618114399379743815107454293, 3.03256482828411403265010572935, 3.51653862370505614187264954230, 4.41692881621028603232660294629, 5.32168130095232201363975590384, 6.34905831088897781344267510925, 7.10388434769505894971710542534, 8.220434958157284188382910396681, 8.700800872763214595582643100438

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