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Properties

Label 2-1350-45.32-c1-0-2
Degree $2$
Conductor $1350$
Sign $-0.144 - 0.989i$
Analytic cond. $10.7798$
Root an. cond. $3.28326$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.448 − 1.67i)7-s + (−0.707 − 0.707i)8-s + (−3 + 1.73i)11-s + (−0.896 − 3.34i)13-s + (−0.866 + 1.50i)14-s + (0.500 + 0.866i)16-s + (−4.24 + 4.24i)17-s + 2i·19-s + (3.34 − 0.896i)22-s + (2.89 − 0.776i)23-s + 3.46i·26-s + (1.22 − 1.22i)28-s + (4.33 + 7.5i)29-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (0.433 + 0.249i)4-s + (0.169 − 0.632i)7-s + (−0.249 − 0.249i)8-s + (−0.904 + 0.522i)11-s + (−0.248 − 0.928i)13-s + (−0.231 + 0.400i)14-s + (0.125 + 0.216i)16-s + (−1.02 + 1.02i)17-s + 0.458i·19-s + (0.713 − 0.191i)22-s + (0.604 − 0.161i)23-s + 0.679i·26-s + (0.231 − 0.231i)28-s + (0.804 + 1.39i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.144 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1350\)    =    \(2 \cdot 3^{3} \cdot 5^{2}\)
Sign: $-0.144 - 0.989i$
Analytic conductor: \(10.7798\)
Root analytic conductor: \(3.28326\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1350} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1350,\ (\ :1/2),\ -0.144 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5722999919\)
\(L(\frac12)\) \(\approx\) \(0.5722999919\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.448 + 1.67i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (3 - 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.896 + 3.34i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.24 - 4.24i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.89 + 0.776i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-4.33 - 7.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5 - 8.66i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.44 + 2.44i)T + 37iT^{2} \)
41 \( 1 + (7.5 + 4.33i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-10.0 - 2.68i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (2.89 + 0.776i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.5 - 11.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (11.7 - 3.13i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (9.79 - 9.79i)T - 73iT^{2} \)
79 \( 1 + (-3.46 + 2i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.776 + 2.89i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + (1.79 - 6.69i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.05173495474713434247451276799, −8.840280951775363961944328944004, −8.388113630187489113058324270611, −7.35466137274138176165190113037, −6.93805999981244774283290733523, −5.66881968207275227120375684728, −4.77460134261617897080358081203, −3.62765131724331748786732799590, −2.56229589224577831168919157328, −1.33698929090662285058329202715, 0.29580562985145656932893037630, 2.09432160616555126721282508934, 2.83155928471805643282997268903, 4.40159895078919866889966590420, 5.28184089959591384553635085417, 6.19599447400907699803129004670, 7.05177200810238452840850663664, 7.82629796389220674051988718807, 8.710655432880915534807821109560, 9.249126757586596953727211920666

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