Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} $
Sign $0.999 + 0.0425i$
Motivic weight 4
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.68 + 1.56i)2-s + (4.25 − 7.93i)3-s + (11.1 + 11.5i)4-s + (1.01 − 1.75i)5-s + (28.0 − 22.5i)6-s + (20.0 − 11.5i)7-s + (22.8 + 59.7i)8-s + (−44.8 − 67.4i)9-s + (6.48 − 4.88i)10-s + (4.32 − 2.49i)11-s + (138. − 39.1i)12-s + (−137. + 238. i)13-s + (91.8 − 11.2i)14-s + (−9.63 − 15.5i)15-s + (−9.35 + 255. i)16-s − 266.·17-s + ⋯
L(s)  = 1  + (0.920 + 0.391i)2-s + (0.472 − 0.881i)3-s + (0.694 + 0.719i)4-s + (0.0406 − 0.0703i)5-s + (0.779 − 0.626i)6-s + (0.408 − 0.236i)7-s + (0.357 + 0.934i)8-s + (−0.553 − 0.832i)9-s + (0.0648 − 0.0488i)10-s + (0.0357 − 0.0206i)11-s + (0.962 − 0.271i)12-s + (−0.815 + 1.41i)13-s + (0.468 − 0.0573i)14-s + (−0.0428 − 0.0690i)15-s + (−0.0365 + 0.999i)16-s − 0.920·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0425i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(36\)    =    \(2^{2} \cdot 3^{2}\)
\( \varepsilon \)  =  $0.999 + 0.0425i$
motivic weight  =  \(4\)
character  :  $\chi_{36} (31, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 36,\ (\ :2),\ 0.999 + 0.0425i)\)
\(L(\frac{5}{2})\)  \(\approx\)  \(2.62068 - 0.0558253i\)
\(L(\frac12)\)  \(\approx\)  \(2.62068 - 0.0558253i\)
\(L(3)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-3.68 - 1.56i)T \)
3 \( 1 + (-4.25 + 7.93i)T \)
good5 \( 1 + (-1.01 + 1.75i)T + (-312.5 - 541. i)T^{2} \)
7 \( 1 + (-20.0 + 11.5i)T + (1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (-4.32 + 2.49i)T + (7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 + (137. - 238. i)T + (-1.42e4 - 2.47e4i)T^{2} \)
17 \( 1 + 266.T + 8.35e4T^{2} \)
19 \( 1 + 367. iT - 1.30e5T^{2} \)
23 \( 1 + (544. + 314. i)T + (1.39e5 + 2.42e5i)T^{2} \)
29 \( 1 + (319. + 553. i)T + (-3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-1.19e3 - 687. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 1.46e3T + 1.87e6T^{2} \)
41 \( 1 + (-593. + 1.02e3i)T + (-1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-1.43e3 + 825. i)T + (1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-307. + 177. i)T + (2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 - 5.29e3T + 7.89e6T^{2} \)
59 \( 1 + (5.22e3 + 3.01e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (833. + 1.44e3i)T + (-6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.90e3 + 1.10e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 524. iT - 2.54e7T^{2} \)
73 \( 1 + 1.49e3T + 2.83e7T^{2} \)
79 \( 1 + (-4.44e3 + 2.56e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-6.91e3 + 3.99e3i)T + (2.37e7 - 4.11e7i)T^{2} \)
89 \( 1 + 8.86e3T + 6.27e7T^{2} \)
97 \( 1 + (3.40e3 + 5.90e3i)T + (-4.42e7 + 7.66e7i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−15.35614863937329353171339086271, −14.24976564972502006955409757769, −13.54434889492076414789939287194, −12.31024922878486779692197615911, −11.31268100914533755000581056275, −8.949582666846477221671529163939, −7.49568194693492799059561559321, −6.45913753963249324005573412058, −4.45975524648245192131640403958, −2.30395528594766952258425361140, 2.63311165925361988311112406932, 4.34082135299595513746320122971, 5.72111514424688024676458147717, 7.953848360679165524871948958154, 9.796645047809313158820662128396, 10.75256568124725415062037193696, 12.10478931841138962939248872155, 13.47133645937089428884729936441, 14.65388405223964074857347241198, 15.28641870864296141823203565756

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