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Properties

Label 2-936-13.10-c1-0-7
Degree $2$
Conductor $936$
Sign $0.113 - 0.993i$
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  + 3.17i·5-s + (2.98 + 1.72i)7-s + (2.98 − 1.72i)11-s + (−0.362 + 3.58i)13-s + (0.886 − 1.53i)17-s + (−2.88 − 1.66i)19-s + (0.193 + 0.334i)23-s − 5.07·25-s + (−2.28 − 3.96i)29-s + 11.0i·31-s + (−5.46 + 9.46i)35-s + (−1.40 + 0.810i)37-s + (1.96 − 1.13i)41-s + (5.36 − 9.29i)43-s + 8.11i·47-s + ⋯
L(s)  = 1  + 1.41i·5-s + (1.12 + 0.650i)7-s + (0.898 − 0.518i)11-s + (−0.100 + 0.994i)13-s + (0.215 − 0.372i)17-s + (−0.661 − 0.382i)19-s + (0.0402 + 0.0697i)23-s − 1.01·25-s + (−0.424 − 0.735i)29-s + 1.98i·31-s + (−0.923 + 1.59i)35-s + (−0.230 + 0.133i)37-s + (0.306 − 0.177i)41-s + (0.818 − 1.41i)43-s + 1.18i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $0.113 - 0.993i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32815 + 1.18524i\)
\(L(\frac12)\) \(\approx\) \(1.32815 + 1.18524i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.362 - 3.58i)T \)
good5 \( 1 - 3.17iT - 5T^{2} \)
7 \( 1 + (-2.98 - 1.72i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.98 + 1.72i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.886 + 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.88 + 1.66i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.193 - 0.334i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.28 + 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 11.0iT - 31T^{2} \)
37 \( 1 + (1.40 - 0.810i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.96 + 1.13i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.36 + 9.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.11iT - 47T^{2} \)
53 \( 1 + 11.6T + 53T^{2} \)
59 \( 1 + (-5.47 - 3.16i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.21 - 2.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.91 - 4.57i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-8.88 - 5.12i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 8.40iT - 73T^{2} \)
79 \( 1 - 8.22T + 79T^{2} \)
83 \( 1 + 1.11iT - 83T^{2} \)
89 \( 1 + (-15.4 + 8.91i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-3.88 - 2.24i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.48571043434686171766030892943, −9.278260007598309960681611898073, −8.696684700529666464184418985990, −7.62526954416004617245991977508, −6.80741496417024262024448236453, −6.13493826142290456674192559392, −4.98619530727778951574013336031, −3.90776265483014493965202979127, −2.77784164491775583936073674375, −1.73415636091201125550560902691, 0.913757996624565677268172144534, 1.89914000987722650852310317962, 3.82090741108355281682473227508, 4.55862795502381454215631626353, 5.29612040790749624512380775922, 6.34211678857356642913064416831, 7.73490215360039171003103441237, 8.036663313104464351825193129251, 9.021072255852790026539413749256, 9.737842356833795413125728239764

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