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Properties

Label 2-936-9.4-c1-0-23
Degree $2$
Conductor $936$
Sign $0.685 + 0.728i$
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  + (0.400 − 1.68i)3-s + (0.618 − 1.07i)5-s + (2.31 + 4.00i)7-s + (−2.67 − 1.34i)9-s + (−0.517 − 0.896i)11-s + (−0.5 + 0.866i)13-s + (−1.55 − 1.47i)15-s + 7.43·17-s + 4.41·19-s + (7.67 − 2.29i)21-s + (2.42 − 4.20i)23-s + (1.73 + 3.00i)25-s + (−3.34 + 3.97i)27-s + (−3.04 − 5.27i)29-s + (3.58 − 6.21i)31-s + ⋯
L(s)  = 1  + (0.231 − 0.972i)3-s + (0.276 − 0.478i)5-s + (0.873 + 1.51i)7-s + (−0.893 − 0.449i)9-s + (−0.156 − 0.270i)11-s + (−0.138 + 0.240i)13-s + (−0.402 − 0.379i)15-s + 1.80·17-s + 1.01·19-s + (1.67 − 0.500i)21-s + (0.506 − 0.876i)23-s + (0.347 + 0.601i)25-s + (−0.643 + 0.765i)27-s + (−0.565 − 0.979i)29-s + (0.644 − 1.11i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.685 + 0.728i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (625, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.685 + 0.728i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84387 - 0.796460i\)
\(L(\frac12)\) \(\approx\) \(1.84387 - 0.796460i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.400 + 1.68i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.618 + 1.07i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.31 - 4.00i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.517 + 0.896i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 7.43T + 17T^{2} \)
19 \( 1 - 4.41T + 19T^{2} \)
23 \( 1 + (-2.42 + 4.20i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.04 + 5.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.58 + 6.21i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.61T + 37T^{2} \)
41 \( 1 + (0.532 - 0.921i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.34 - 2.32i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.00 - 6.93i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 10.4T + 53T^{2} \)
59 \( 1 + (-4.04 + 7.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.92 + 10.2i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.406 - 0.704i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.3T + 71T^{2} \)
73 \( 1 - 8.15T + 73T^{2} \)
79 \( 1 + (3.78 + 6.56i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.56 - 4.43i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.37T + 89T^{2} \)
97 \( 1 + (-4.96 - 8.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.571570782919654970824445130346, −9.100754665464119383564137695947, −8.038123855691200513514566633388, −7.81454005574319018582241045623, −6.41536175902780847638777167605, −5.55682677391349221629023196091, −5.05371434586894023977901111007, −3.24097700987931040875767830028, −2.25102661732434490705354223954, −1.17865578965765771464023408264, 1.31976097712711581506878659429, 3.08643582022672031408837252398, 3.75618952385400429417760536385, 4.95657563134325654807942811453, 5.45988587045614439571412169650, 7.04801906738183729107983817418, 7.59270102592239172281757304945, 8.481904087716893136533345421644, 9.601276813616637707698682664452, 10.35923425060268747342139132159

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