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Properties

Label 2-936-39.11-c1-0-2
Degree $2$
Conductor $936$
Sign $-0.741 - 0.670i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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  + (−1.79 + 1.79i)5-s + (1.02 + 3.83i)7-s + (1.05 − 3.95i)11-s + (1.54 + 3.25i)13-s + (1.68 − 2.91i)17-s + (−7.63 + 2.04i)19-s + (1.54 + 2.67i)23-s − 1.41i·25-s + (−7.02 + 4.05i)29-s + (0.618 + 0.618i)31-s + (−8.70 − 5.02i)35-s + (−8.92 − 2.39i)37-s + (−6.25 − 1.67i)41-s + (8.40 + 4.85i)43-s + (−4.37 − 4.37i)47-s + ⋯
L(s)  = 1  + (−0.801 + 0.801i)5-s + (0.388 + 1.44i)7-s + (0.319 − 1.19i)11-s + (0.428 + 0.903i)13-s + (0.408 − 0.707i)17-s + (−1.75 + 0.469i)19-s + (0.322 + 0.558i)23-s − 0.283i·25-s + (−1.30 + 0.753i)29-s + (0.111 + 0.111i)31-s + (−1.47 − 0.849i)35-s + (−1.46 − 0.392i)37-s + (−0.977 − 0.261i)41-s + (1.28 + 0.740i)43-s + (−0.637 − 0.637i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.741 - 0.670i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.741 - 0.670i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.348128 + 0.903530i\)
\(L(\frac12)\) \(\approx\) \(0.348128 + 0.903530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
13 \( 1 + (-1.54 - 3.25i)T \)
good5 \( 1 + (1.79 - 1.79i)T - 5iT^{2} \)
7 \( 1 + (-1.02 - 3.83i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-1.05 + 3.95i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.63 - 2.04i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.54 - 2.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.02 - 4.05i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.618 - 0.618i)T + 31iT^{2} \)
37 \( 1 + (8.92 + 2.39i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (6.25 + 1.67i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-8.40 - 4.85i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.37 + 4.37i)T + 47iT^{2} \)
53 \( 1 - 13.8iT - 53T^{2} \)
59 \( 1 + (-4.03 + 1.07i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (4.06 - 7.04i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.17 + 11.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (0.166 + 0.622i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (0.788 - 0.788i)T - 73iT^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + (0.0917 - 0.0917i)T - 83iT^{2} \)
89 \( 1 + (3.46 - 12.9i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-6.54 + 1.75i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.74513974377557577202365414135, −9.211901354898195561891726977011, −8.801829370349335409038165268251, −7.957995140916279522610249835408, −6.94408874627828536093554645886, −6.09980833043255330775814760084, −5.27472600223900550694240464912, −3.92596618385649394068416768098, −3.11672193062360462970470291728, −1.87913193452879943882823421960, 0.45266864731144251107474444228, 1.81881203101676438935566950469, 3.73016761455899966548572656429, 4.26599766876958418708150808893, 5.07199824754716557258195332081, 6.48433571168777844625766789744, 7.31532087928066485096598002110, 8.101468165449177755155623632927, 8.654957957769500940042322858396, 9.912179183693108680135248115401

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