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Properties

Label 2-1950-13.3-c1-0-33
Degree $2$
Conductor $1950$
Sign $0.794 - 0.607i$
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 + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.499 + 0.866i)6-s + (1.78 − 3.08i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (2.06 + 3.57i)11-s − 0.999·12-s + (3.34 − 1.35i)13-s + 3.56·14-s + (−0.5 − 0.866i)16-s + (2.56 − 4.43i)17-s − 0.999·18-s + (1.78 − 3.08i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.204 + 0.353i)6-s + (0.673 − 1.16i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.621 + 1.07i)11-s − 0.288·12-s + (0.926 − 0.375i)13-s + 0.951·14-s + (−0.125 − 0.216i)16-s + (0.621 − 1.07i)17-s − 0.235·18-s + (0.408 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1950\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.794 - 0.607i$
Analytic conductor: \(15.5708\)
Root analytic conductor: \(3.94598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1950} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1950,\ (\ :1/2),\ 0.794 - 0.607i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.659701135\)
\(L(\frac12)\) \(\approx\) \(2.659701135\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-3.34 + 1.35i)T \)
good7 \( 1 + (-1.78 + 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.06 - 3.57i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.56 + 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.78 + 3.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.84 + 6.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.28 + 5.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.68T + 31T^{2} \)
37 \( 1 + (-2.06 - 3.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.12 - 3.67i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-2.28 + 3.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + 4.43T + 53T^{2} \)
59 \( 1 + (5.28 - 9.14i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.12 - 12.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.43 + 4.22i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3T + 73T^{2} \)
79 \( 1 - 7.43T + 79T^{2} \)
83 \( 1 + 1.12T + 83T^{2} \)
89 \( 1 + (0.903 + 1.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.561 + 0.972i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.299798214976689710472462398622, −8.228794583378950439620842344270, −7.74990887223061392611039570598, −6.94892674098299596603020845599, −6.16811977613832819162866277202, −4.97401820693630656476477889974, −4.42834041583236543987221232228, −3.76474908758081342739552066432, −2.56809119830715925666651768577, −0.979687609296254591784940691191, 1.32785401268334758728197865330, 1.93544158199276962788318377983, 3.32331371544325126522387691653, 3.76915656329082789458868691119, 5.18842115578722503515741826327, 5.92005910430781055589221638894, 6.38084831624460273275557987701, 7.920983976234376704477325389338, 8.272401944892662011298026777447, 9.124112982967310450982164833589

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