Properties

Label 2-684-19.12-c4-0-25
Degree $2$
Conductor $684$
Sign $0.463 + 0.886i$
Analytic cond. $70.7050$
Root an. cond. $8.40862$
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  + (1.89 + 3.27i)5-s + 36.8·7-s + 226.·11-s + (−177. − 102. i)13-s + (−15.7 − 27.2i)17-s + (−322. − 162. i)19-s + (439. − 761. i)23-s + (305. − 528. i)25-s + (−203. − 117. i)29-s + 1.65e3i·31-s + (69.6 + 120. i)35-s − 1.06e3i·37-s + (−70.5 + 40.7i)41-s + (617. + 1.07e3i)43-s + (−1.14e3 + 1.99e3i)47-s + ⋯
L(s)  = 1  + (0.0756 + 0.131i)5-s + 0.751·7-s + 1.87·11-s + (−1.05 − 0.606i)13-s + (−0.0544 − 0.0942i)17-s + (−0.892 − 0.450i)19-s + (0.831 − 1.44i)23-s + (0.488 − 0.846i)25-s + (−0.241 − 0.139i)29-s + 1.71i·31-s + (0.0568 + 0.0985i)35-s − 0.779i·37-s + (−0.0419 + 0.0242i)41-s + (0.334 + 0.578i)43-s + (−0.520 + 0.901i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(70.7050\)
Root analytic conductor: \(8.40862\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :2),\ 0.463 + 0.886i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.300441489\)
\(L(\frac12)\) \(\approx\) \(2.300441489\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 + (322. + 162. i)T \)
good5 \( 1 + (-1.89 - 3.27i)T + (-312.5 + 541. i)T^{2} \)
7 \( 1 - 36.8T + 2.40e3T^{2} \)
11 \( 1 - 226.T + 1.46e4T^{2} \)
13 \( 1 + (177. + 102. i)T + (1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + (15.7 + 27.2i)T + (-4.17e4 + 7.23e4i)T^{2} \)
23 \( 1 + (-439. + 761. i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (203. + 117. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 - 1.65e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.06e3iT - 1.87e6T^{2} \)
41 \( 1 + (70.5 - 40.7i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-617. - 1.07e3i)T + (-1.70e6 + 2.96e6i)T^{2} \)
47 \( 1 + (1.14e3 - 1.99e3i)T + (-2.43e6 - 4.22e6i)T^{2} \)
53 \( 1 + (2.87e3 + 1.65e3i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (-1.36e3 + 786. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-1.68e3 + 2.92e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-2.80e3 - 1.61e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + (-7.88e3 + 4.55e3i)T + (1.27e7 - 2.20e7i)T^{2} \)
73 \( 1 + (3.14e3 + 5.45e3i)T + (-1.41e7 + 2.45e7i)T^{2} \)
79 \( 1 + (705. - 407. i)T + (1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 5.75e3T + 4.74e7T^{2} \)
89 \( 1 + (1.01e4 + 5.86e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + (8.99e3 - 5.19e3i)T + (4.42e7 - 7.66e7i)T^{2} \)
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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

−9.681771425961829498811752490757, −8.856777710464977063433923611428, −8.134014980949540242104211903602, −6.90781886556589359356168304502, −6.42705143231283088605586124903, −4.98220090393953103586402624263, −4.36607732294515406218547934006, −3.01598075310876759801316285208, −1.82382169917767980230391505293, −0.58944359582818199438946332258, 1.18619319516607933485755042952, 2.05050349476841118494298397660, 3.65324120722103438271095771605, 4.48389807972903258836864660487, 5.48802780056102073878106445605, 6.61772081610533894466397739061, 7.33121592731182715013152231001, 8.400477409332847247677214467884, 9.282322087304579365351439861382, 9.784245362511057406058667017852

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