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Properties

Label 2-304-19.12-c4-0-23
Degree $2$
Conductor $304$
Sign $0.827 - 0.560i$
Analytic cond. $31.4244$
Root an. cond. $5.60575$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.81 − 4.50i)3-s + (15.3 + 26.6i)5-s + 70.8·7-s + (0.166 − 0.287i)9-s + 6.58·11-s + (73.7 + 42.6i)13-s + (240. + 138. i)15-s + (41.8 + 72.4i)17-s + (−125. + 338. i)19-s + (553. − 319. i)21-s + (238. − 413. i)23-s + (−161. + 279. i)25-s + 727. i·27-s + (−1.40e3 − 813. i)29-s + 1.59e3i·31-s + ⋯
L(s)  = 1  + (0.867 − 0.501i)3-s + (0.615 + 1.06i)5-s + 1.44·7-s + (0.00205 − 0.00355i)9-s + 0.0544·11-s + (0.436 + 0.252i)13-s + (1.06 + 0.617i)15-s + (0.144 + 0.250i)17-s + (−0.346 + 0.938i)19-s + (1.25 − 0.724i)21-s + (0.451 − 0.781i)23-s + (−0.258 + 0.447i)25-s + 0.997i·27-s + (−1.67 − 0.966i)29-s + 1.66i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(31.4244\)
Root analytic conductor: \(5.60575\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :2),\ 0.827 - 0.560i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.519898504\)
\(L(\frac12)\) \(\approx\) \(3.519898504\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (125. - 338. i)T \)
good3 \( 1 + (-7.81 + 4.50i)T + (40.5 - 70.1i)T^{2} \)
5 \( 1 + (-15.3 - 26.6i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 70.8T + 2.40e3T^{2} \)
11 \( 1 - 6.58T + 1.46e4T^{2} \)
13 \( 1 + (-73.7 - 42.6i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (-41.8 - 72.4i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-238. + 413. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (1.40e3 + 813. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.59e3iT - 9.23e5T^{2} \)
37 \( 1 + 593. iT - 1.87e6T^{2} \)
41 \( 1 + (261. - 151. i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (222. + 386. i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (-119. + 206. i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (-2.94e3 - 1.70e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-4.73e3 + 2.73e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-789. + 1.36e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (4.14e3 + 2.39e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-2.42e3 + 1.40e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.44e3 + 5.96e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (-2.53e3 + 1.46e3i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 1.20e4T + 4.74e7T^{2} \)
89 \( 1 + (-9.80e3 - 5.65e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (-1.45e3 + 837. i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−10.94770561310921986382969283032, −10.47390882090484021915475491399, −9.066733706110600264529566752614, −8.214648353935470535743337214877, −7.47478245048085853899139538828, −6.40466947841486868746180120929, −5.20836619784438790377671100161, −3.68658967249312269770905175205, −2.34267479354852341059685744702, −1.62729362461874275049592287335, 1.05103993683448098634153625437, 2.24283443003063878670407024744, 3.80525998670359297106486486913, 4.88832894000798342433853989865, 5.68206182163790893025033524055, 7.41525879004491912653113906473, 8.463213743537093224514205457174, 8.975279412680217408733645435073, 9.748186790188616388326631349084, 11.04624626981145176234810446276

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